NIST Calibration Services for Hydrometers 

NIST Special Publication 250-78 

John. D. Wright, Vern E. Bean, Jesus Aguilera 

April 19, 2010 

Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratory 

National Institute of Standards and Technology 

Gaithersburg, Maryland, 20899
Hydrometer Calibration Service NIST SP 250-78

Table of Contents Page 

1. Description of Service. ..................................................................................................................1 

2. Introduction ....................................................................................................................................1 

3. Cuckow’s Method ..........................................................................................................................4 

4. Procedures and Apparatus .............................................................................................................9 

5. Surface Positioning Process .........................................................................................................10 

6. Surface Tension and Contact Angle Effects ................................................................................11

7. Uncertainty of the NIST Hydrometer Calibration .......................................................................12 

8. Uncertainty When the Hydrometer Is Used .................................................................................16 

9. Validation with Liquids of Known Density .................................................................................18

10. Comparison to NIST Historical Results for a Reference Hydrometer ......................................19 

11. Summary ....................................................................................................................................20 

Appendix A. Corrections to Hydrometer Readings for Surface Tension Effects ............................22 

Appendix B: Corrections to Hydrometers for Temperature Effects ................................................27 

Appendix C. Conversion of Liquid Density to Other Hydrometer Scales .....................................31 

Appendix D. Nomenclature .............................................................................................................32 

Appendix E. Sample Calibration Report..................................................................................1 
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1. Description of Service. 

The National Institute of Standards and Technology (NIST) provides calibration services for 

hydrometers used to measure liquid density. Hydrometers with scales for density, specific 

gravity, proof spirit for alcohol solutions, API degrees for petroleum measurements, degrees 

Baume heavy and degrees Baume light, and others are accepted for calibration. NIST 

calibrations are done by weighing the hydrometer while it is immersed to specified scale 

markings in a liquid of known density (tridecane). Normally a calibration is done at three points, 

at approximately 10 %, 50 %, and 90 % of the scale. The calibration report includes a simple 

equation for each calibration point that gives the user a correction to the hydrometer reading that 

accounts for the surface tension of the liquid in which the hydrometer is used. The calibration 

results have an expanded uncertainty (with coverage factor of 2) of 100 parts in 106

 or less of the 

liquid density. The apparatus was validated by comparisons using liquids of known density: 

water, toluene, tridecane, and trichloroethylene. Results from these comparisons agreed within 

40 parts in 106 or less (see Table 3). The new calibration method is consistent with earlier, 

manual calibrations performed by NIST using reference hydrometers (see Figure 5). 

Calibration customers should consult the web address 

http://ts.nist.gov/MeasurementServices/Calibrations/mechanical_index.cfm to find the most 

current information regarding our calibration service, calibration fees, technical contacts, turn 

around times, and instrument submittal procedures. The instructions for domestic customers have 

the sub-headings: A) Customer Inquiries, 

B) Pre-arrangements and Scheduling, 

C) Purchase Orders, 

D) Shipping, Insurance, and Risk of Loss, 

E) Turnaround Time 

F) Customer Checklist. 

There are also special instructions for foreign customers. 

2. Introduction 

A hydrometer is a simple and inexpensive instrument for measuring the density of liquids. The 

typical hydrometer... “is a glass instrument consisting of a bulb weighted at the lower end and 

surmounted by a stem, small in diameter compared with the bulb, in which is enclosed a scale on 

which readings are taken”[1]. The instrument floats vertically in the liquid and the liquid density 

is measured by visually reading where the plane of the free surface of the liquid intersects the 

hydrometer scale (see Figure 1). Numerous standards documents are available that specify 

design and usage of hydrometers [2, 3, 4, 5]. 

Figure 1 illustrates the forces acting on a freely floating hydrometer. Two buoyant forces act in 

the upward direction, one due to the volume of the bulb and the portion of the stem below the 

 

[1] Stott V 1938 Hydrometers, in The Science of Petroleum, ed A E Dunstan (London, Oxford University 

Press) vol III p 2322-7 

[2] ISO 649-1 1981 Laboratory Glassware – Density Hydrometers for General Purposes – Specification. 

[3] ISO 649-2 1981 Laboratory Glassware - Density Hydrometers for General Purposes – Test Methods 

and Use.

[4] ASTM E-100 2005 Standard Specification for ASTM Hydrometers, ASTM Standards 14.03. 

[5] ASTM E-126 2005 Standard Test Method for Inspection and Verification of Hydrometers, ASTM 

Standards 14.03. 
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liquid surface that are displacing liquid, the other due to the portion of the stem that is displacing 

air. Gravity acts in the downward direction. In the hydrometer’s equilibrium position, the upward 

and downward forces are balanced. The equilibrium position is mainly dependent on the density 

of the liquid. Hydrometer manufacturers position scales inside the hydrometer stem so that users 

can read the liquid density from the surface-scale intersection. Inevitably, there are small errors 

in the positioning of the scale and the density measurements can be improved by adding 

corrections to the hydrometer readings. This document describes the calibration procedure that 

applies Cuckow’s method [6] to determine these hydrometer corrections. 

Forces due to the liquid-vapor surface tension and the contact angle between the liquid-vapor 

interface and the hydrometer stem complicate hydrometer density measurements (see Figure 1). 

Most liquids partially or completely wet clean glass. For such liquids, a curved liquid-vapor 

interface (meniscus) rises up the hydrometer stem and, in effect, pulls the hydrometer deeper into 

the liquid. (A few liquids, e.g. mercury, do not wet clean glass. For such liquids, the meniscus is 

depressed below the free surface and surface tension pulls the hydrometer upwards.) When 

aqueous solutions contact a, clean, glass hydrometer, the rising meniscus can alter the 

hydrometer’s equilibrium position by as much as a 0.2 % reduction of the liquid’s density. For 

hydrocarbon liquids the reduction of the apparent density can be as large as 0.1%. Hence, for 

most hydrometer applications, it is necessary to account for surface tension effects in order to 

obtain the desired density accuracy. The NIST hydrometer calibration service provides 

hydrometer corrections that are a function of the surface tension of the customer’s liquid 

assuming that the contact angle between the meniscus and the stem is zero. [see Section 3, 

Equation (10).] Appendix D is a sample calibration report. 

Figure 1. Schematic of a freely floating hydrometer with representations of the forces imposed on it. 

See Section 3 for nomenclature. 

 

[6] Cuckow F W 1949 A New Method of High Accuracy for the Calibration of Reference Standard 

Hydrometers, J. Soc. Chem. Industry, 68, 44-9 
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Although hydrometers with SI units of g/cm3 are desirable today, for historical reasons a number 

of other hydrometer units prevail. In the United States, the most common unit for hydrometers is 

specific gravity, the density of the measured liquid at a reference temperature relative to the 

density of distilled water at a reference temperature (i.e. specific gravity 60 °F / 60 °F+). This 

unit was developed because distilled water was a convenient reference material. 

Hydrometers can be used to measure any liquid property that is a function of density simply by 

putting an appropriate scale in the hydrometer stem. Examples are percentage of ethyl alcohol, 

sugar, or sulfuric acid in water. Other scales include the Baume scale used by the sugar industry, 

and the American Petroleum Institute or API scale used by the petroleum industry. Conversions 

between the various scales are discussed in the literature [7] and in Appendix A of this 

document. 

There are two primary methods for calibrating hydrometers. One method requires floating the 

hydrometer in several liquids of different densities, which are determined by applying 

Archimedes’ principle to a sinker of known mass and volume. The second method (Cuckow’s 

method) requires hydrostatic weighing, i.e. using a balance to measure the apparent mass∗ of the 

hydrometer while it is immersed to several levels in a single liquid of known density. Cuckow’s 

method has many advantages. It eliminates the inconvenience of storing a collection of liquids, 

such as aqueous solutions of acids and mixtures of volatile hydrocarbons. It permits the 

calibration of a hydrometer at specified scale marks without the inconvenience of preparing 

liquid mixtures to specified densities. In addition, Cuckow’s method can be automated more 

easily. In the apparatus described here, we use tridecane as the liquid of known density because 

of six attributes: 

1) commercially available in high purity, 

2) a stable density that is low enough to calibrate most hydrometers without a sinker, 

3) low vapor pressure, 

4) non-hydroscopic (i.e., does not absorb water vapor from air), 

5) a low, stable surface tension in contact with air, 

6) low toxicity. 

National standards laboratories are interested in automating the calibration of hydrometers in 

order to reduce uncertainties and reduce the labor involved thereby reducing cost to the customer 

[8, 9, 10]. The Fluid Metrology Group of NIST designed and built an automated, primary 

standard apparatus for the calibration of hydrometers using Cuckow’s method and began using it 

for customer calibrations in 2007. The NIST apparatus suspends the customer’s hydrometer from 

a balance and immerses it far enough into tridecane so that the hydrometer mark to be calibrated 

 

+ specific gravity 60 °F / 60 °F is not an SI unit, but is still a widely found on hydrometers used in the 

United States.

[7] Hughes J C 1954 Testing of Hydrometers, National Bureau of Standards Circular 555, (Gaithersburg, 

National Institute of Standards and Technology) 

∗

 Apparent mass is force divided by gravitational acceleration as measure by a balance. 

[8] Jacques C 1999 Calibration of Hydrometers – Procedures, NRCC 42747, (Canada, National Research 

Council Canada) 

[9] Lee Y J, Chang K H, and Oh C Y 2004 Automatic Alignment Method for Calibration of Hydrometers, 

Metrologia, 41, S100-S104 

[10] Lorefice S and Malengo A 2006 Calibration of Hydrometers, Meas. Sci. Technol., 17, 2560-6 
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is aligned with the free surface. The apparent mass of the hydrometer in this configuration is 

combined with its apparent mass in air to calculate the correction to the hydrometer scale as 

shown in Section 3. The automated NIST apparatus is shown schematically in Figure 2. The 

apparatus positions a horizontal laser light sheet on the hydrometer mark to be calibrated. Then it 

raises or lowers the thermostatted water bath until the free surface of the tridecane is aligned 

with the light sheet. A laser power meter indicates the correct alignment by a characteristic 

change in its reading. (See Section 5.) 

The NIST apparatus achieves an expanded (k = 2) uncertainty of 100 parts in 106

 or less for the 

calibration results. Customers can expect larger uncertainties when they use their hydrometer for 

liquid density measurements due to the effects of surface tension, contact angle, improper 

reading of the meniscus, and temperature. 

 

Figure 2. A schematic diagram of the NIST apparatus that applies Cuckow’s method. 

3. Cuckow’s Method 

The development of the data reduction equation presented here follows the work of Cuckow [6]. 

The derivation uses the force balance equations for the hydrometer in three circumstances: 1, 

suspended from a balance in air, 2. freely floating at a particular mark in a liquid of known 

surface tension and unknown density (e.g. the customer’s liquid), and 3, suspended at the same 

mark in another liquid of known density and surface tension (e.g. tridecane at NIST). The force 

balance equations are combined with the assumption that the density of the surrounding air is 

equal in all three cases to obtain corrections for any calibration mark on the hydrometer scale. 

The hydrometer is considered to be an instrument to measure liquid density, marked in units of 

g/cm3

, but corrections for other hydrometer scales can be determined as well. 

We write an equation for the forces acting on a hydrometer for three cases: 

1. floating freely in a liquid of density ρ L at reference temperature T0 , 

2. suspended from a balance in air, and 
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3. suspended from a balance while immersed to the identical level as in Case 1, but in a 

liquid of known density ρ T at temperature TT . (Tridecane is the liquid of known 

density, hence the subscript “T”.) Note that TT need not be equal to the hydrometer 

reference temperatureT0 . 

Solving the three equations simultaneously yields the data reduction equation. 

Freely Floating: Assuming a rising meniscus, there are two downward forces acting on a 

hydrometer floating freely in a liquid: the force due to the surface tension of the liquid acting on 

the stem of the hydrometer and the force due to gravity. There are also two upward forces: the 

buoyant force due to the liquid displaced by the volume of the hydrometer below the liquid 

surface and the buoyant force of the air displaced by the volume of the hydrometer stem above 

the liquid surface. Equating the sum of upward forces to the sum of the downward forces, we 

have 

L L L a1 mg + π Dγ cosθ = gVρ + gvρ (1)

where: 

m : mass of the hydrometer, 

g : local acceleration due to gravity, 

D : diameter of the hydrometer stem at the surface of the liquid, 

γL : surface tension of the liquid in which the hydrometer is used, 

θ L : contact angle of the liquid on the hydrometer stem, 

V : volume of the hydrometer below the liquid surface, 

ρ L : density of the liquid, 

v : volume of the hydrometer stem above the liquid surface, and 

ρ a1 : density of the air at the time the hydrometer scale was read. 

For convenience, we introduce the symbol L LL Γ ≡ πγ θ D cos for the surface tension force on the 

stem from the liquid. 

Suspended from a Balance in Air: There are two forces acting on a hydrometer when it is 

suspended in air: the downward force due to gravity and the upward buoyant force due to the air 

displaced by the volume of the hydrometer. The reading of the balance, multiplied by g, is the 

sum of these two forces, 

[ ] 2 1 2 0 a2 gαO = mg − g(V + v ) + β (T −T ) ρ (2)

where: 

α : calibration coefficient for the balance, 

O2 : balance reading, 

β : volumetric thermal expansion coefficient of the glass from which the hydrometer is 

made, 

T2 : air temperature at the time of weighing in air, 

T0 : reference temperature (often it is15.556 °C or 60 °F), and 

ρ a2 : air density at the time of weighing. 

Suspended from a Balance While Partially Immersed: When the hydrometer is suspended while 

immersed in tridecane to the identical level as in the freely floating case, there are again two 
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downward forces: that due to gravity and that due to surface tension acting on the stem of the 

hydrometer. There are also two upward forces: the buoyant force due to the tridecane displaced 

by the volume V and the buoyant force due to the volume of air displaced by the volume v. The 

balance reading, multiplied by g, is the sum of all of these forces: 

[ ] 3 0 a3 1 T cos gαO3 = mg + πDγ T θ T − gVρ + β(T − T ) − gvρ (3)

where: 

O3 : balance reading, 

γT : surface tension of tridecane, 

θ T : contact angle of the tridecane on the hydrometer stem, 

TT : temperature of the tridecane, 

ρ T : density of the tridecane, and 

ρ a3 : density of the air at the time of the hydrostatic weighing in tridecane. 

The buoyant force for the portion of the hydrometer stem in air [ ( ) gvρ a3β TT −T0 ] is negligibly 

small and has been dropped from Equation (3). For convenience, we introduce the symbol 

T TT Γ πγ θ ≡ D cos for the surface tension force from tridecane. 

Cuckow wrote: “The terms ρ a1 v , ρ a2 v , and ρ a3 v in the three Equations (1), (2), and (3) 

represent the air buoyancies acting on the emergent stems of the hydrometer at various times. 

Variations in the density of the air surrounding the stems of hydrometers are neglected in normal 

hydrometry, the resultant changes in buoyant up thrust being too small to affect the hydrometer 

reading significantly. This procedure is equivalent to regarding terms of the form ρ an v as equal 

to a constant, say ‘ k ’”[6]. Thus Cuckow justifies the simplifying assumption, 

v = v = v = k ρ a1 ρ a2 ρ a3 (4)

We checked this approximation by deriving the data reduction equation without making the 

simplifying assumption. We did assume ρ a2 = ρ a3 because both are measured during the 

hydrometer calibration. These measurements are separated by, at most, a few hours which 

normally results in negligibly small changes in air density. However, ρ a1 will be measured by 

the hydrometer user at some other time and place, thus it might differ from ρ a2 and ρ a3 by as 

much as 20 %, the reduction in air density from NIST (altitude 100 m) to an altitude of 2200 m 

[11]. Under these assumptions, we found Cuckow’s approximation [Equation (4)] makes an error 

of less than 7 parts in 106

, which is negligible for most hydrometer measurements. 

Data Reduction Equations: The steps to derive the data reduction equation are: 

1. Substitute k for ρ a1 v , ρ a2 v , and ρ a3 v in Equations (1), (2), and (3). 

2. Solve Equation 1 for ρ L . The result is, 

 

[11] The altitude of Los Alamos National Laboratory is 2200 m. Variations of air density as large as 3 % 

result from weather. Jaeger, K. B., and Davis, R. S., A Primer for Mass Metrology, National Bureau of 

Standards (U.S.), NBS Special Publication 700-1, Industrial Measurement Series, 1984, 79 p. 
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L

L

1 m k

V g

Γ

ρ

⎛ ⎞

= −+ ⎜ ⎟ ⎝ ⎠ (5)

3. Solve Equation (2) for m − k and substitute it into Equation (5). The results are, 

mk O V T T −= + + − α ρβ 2 a2 2 0 [1( )] (6)

L

L 2 a2

1 O

V g

Γ

ρ α ρ

⎛ ⎞

= ++ ⎜ ⎟ ⎝ ⎠ (7)

where (T T ) ρ a2β 2 − 0 is negligibly small and is not used. 

4. Solve Equation (3) for m − k , substitute the result into Equation (6) and solve the 

resulting equation for V , 

( )

[ ]

23 T

T T 0 a2 1( )

OO g V

T T

α Γ

ρ β ρ

− + = + −− (8)

where (T T ) ρ a2β T − 0 is negligibly small and has been dropped. 

5. Substitute Equation (8) into Equation (7) to obtain the final result, 

{ [ ] }( )

( )

T T 0 a2 2 L

L a2

23 T

1( ) TT O g

OO g

ρ β ρα Γ

ρ ρ α Γ

+ −− +

= +

− + (9)

When calibrating a hydrometer for use in liquids that are less dense than tridecane, a stainless 

steel weight is added to cause the hydrometer to sink. The weight is a ring of known mass and 

volume placed over the stem, resting on the top of the bulb. It is submerged in the tridecane 

during the hydrometer calibration. When the weight is used, Equation (3) must be modified to 

include the two forces acting on it: the force due to gravity and the buoyant force due to the 

tridecane it displaced. The modified equation is denoted as Equation (3’). 

{ } sT 0 1 ( ) [1 ( )] 3 s T T 0 s T a3 g O mg m g g V T T V T T gv α =+ + − + − − + − − Γ β β ρρ ⎡ ⎤ ⎣ ⎦

 (3’) 

where: 

ms : mass of the stainless steel weight 

Vs : volume of the stainless steel weight, and 

βs : volumetric thermal expansion coefficient of the stainless steel weight. 

The data reduction equation when the stainless steel weight is used is, 

{ T T 0 a2 2 L [ ] }( )

L a2

2 3 T s sT s T 0

1( )

[1 ( )]

TT O g

O O gm V T T

ρ β ρα Γ

ρ ρ ααΓ ρβ

+ −− +

= +

− + +− + −

(9’)

The hydrometers most commonly calibrated at NIST are designed to measure specific gravity of 

a liquid at 60 °F referenced to water at 60 °F (a sp gr 60 °F / 60 °F hydrometer). For this 

situation, T0 in Equation (9) or (9’) is set to 15.556 °C (60 °F) and ρ L is divided by the density 

of distilled water at T0, 0.999016 g/cm3

 to obtain the true scale reading. 
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Equations (9) and (9’) include forces due to surface tension, those due to the calibration liquid 

(tridecane) as well as those due to the liquid in which the hydrometer is used by the customer. 

For liquids that wet glass, surface tension causes a meniscus to rise on the stem of a hydrometer 

and the force of the meniscus pulls the hydrometer deeper into the liquid, causing the hydrometer 

reading to be too low. Typically, surface tension forces on hydrometers effect density 

measurements by about 0.2 % for aqueous solutions and about 0.1 % for hydrocarbon liquids. 

The hydrometer thermal expansion term in Equations (9) and (9’) correct for differences in 

temperature between the calibration at NIST and the reference temperature of the hydrometer 

(the temperature at which the hydrometer is designed to be used). 

NIST’s balance is calibrated with stainless steel masses and it has an internal algorithm that 

assumes a nominal air density ρa2 so that the balance indicates a buoyancy-corrected mass. The 

force balances used to derive Equation (1) use the apparent mass (i.e., the indication of an ideal 

balance before buoyancy corrections are made). To obtain the apparent mass, the NIST balance 

readings O2 and O3 are multiplied by ( ) α = 1− ρ a2 ρ s = 0.99985 where ρ s is the density of 

stainless steel calibration masses. 

If desired by a calibration customer, NIST will report a value of, expressed in units specified, 

for each hydrometer scale mark that was calibrated where the value of ρL is based on a 

customer-provided value for the surface tension γL and the assumption that the contact angle θL

of the customer’s liquid is zero when the calibrated hydrometer is used. Since the value of the 

customer’s surface tension is not always known at the time of NIST’s calibration (perhaps 

because the hydrometer will be applied in several liquids with different surface tensions) the 

result of the calibration is also reported as an equation that gives a correction to be added to each 

calibrated scale mark: 

 L L L C = A + Bγ cosθ (10) 

where 

 T T 0 a2 2

a2

2 3 T s sT 0 T

{ [1 ( )] }

[1 ( )] s

TT O A R

O O gm V T T

ρ β ρα ρ ααΓ β ρ+ −−

= +−

− + +− + −

 (11) 

where R is the hydrometer reading at the calibrated scale mark and 

 O g

D B A R

2

a2 ( )

α

π = + − ρ . (12) 

 

NIST calibration customers can use Equation (10) to calculate hydrometer corrections for a 

particular liquid under test. The values of A and B are the output of a NIST calibration. (See 

Appendix D for a sample report). In Section 7 of this paper, we will give the uncertainty 

contributed to CL via A and B. In Section 8 of this paper we will analyze the uncertainties that 

arise during usage of a hydrometer by a NIST calibration customer. For example, the values of 

surface tension and contact angle in Equation (10) depend on the liquid under test and the 

cleanliness of the hydrometer when it is used and are not included in the NIST calibration 

uncertainty. Surface tension effects are discussed in Section 6 and in Appendix B. 
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4. Procedures and Apparatus 

Each hydrometer is cleaned by a thorough rinse with ethanol and it is dried with a clean, paper 

towel before it is calibrated in tridecane by Cuckow’s method. 

 

Hydrometer calibration by Cuckow’s method requires that the hydrometer be weighed in air and 

weighed while immersed to the selected scale marks in a liquid of known density, in our case, 

tridecane. In brief, the automated Cuckow’s method procedures are: 

1. Hang the hydrometer from a hook on the underside of the pan of the balance so that the 

stem intersects the free surface of the tridecane. 

2. With the tridecane surface below the scale mark to be calibrated, center a horizontal sheet 

of laser light on that selected scale mark. 

3. Raise the tridecane surface 3 mm above the scale mark and then lower it until the laser 

power meter mounted opposite the laser indicates a sharp decrease in the laser light 

intensity due to the tridecane surface being in the same plane as the scale mark to be 

calibrated (the “surface positioning process” shown in Figures 3 and 4). 

4. Stop the travel of the tridecane surface, pause for the balance readings to stabilize, and 

record the tridecane temperature, the reading of the balance, and the reading of the 

encoder that monitors the travel of the tridecane surface. 

The weighing of the hydrometer suspended in tridecane is done with an analytical balance 

located on top of a cabinet of square cross section, approximately 0.5 m on an edge and 1.5 m 

high (see Figure 2). The balance has a capacity of 205 g and a resolution of 0.0001 g. The 

hydrometer is hung by a chain from the underside of the pan of the balance, which is centered 

over a hole in the top plate of the cabinet so as to be directly above the 100 mm diameter tube 

containing the tridecane. This tube is mounted vertically in a cylindrical, temperature controlled 

bath, which is mounted on a vertical stage such that the bath-and-tube can be raised and lowered 

by a motor under computer control. A linear encoder (5 μm resolution) is attached to the vertical 

stage to monitor the position of the tridecane surface. Although the linear encoder measurements 

are not needed for density calculations, they are useful for calculating the repeatability of the 

surface positioning process. An external bath pumps temperature controlled water through the 

cylinder surrounding the tube of tridecane. A thermometer well mounted within the tridecane 

supports a calibrated thermistor to measure the temperature. 

A CCD camera provides an image of the hydrometer scale on the computer monitor upon which 

cursors are placed to indicate the scale marks to be calibrated. A self-leveling class II laser (a 

laser diode mounted on a magnetically dampened pendulum) is located on its own vertical stage 

with a motor under computer control. The thickness of the laser light sheet (0.2 mm) is defined 

by a 50 mm long optical slit with a polarizing filter, located between the laser and the bath-andtube. A laser power meter is mounted on the opposite side of the bath-and-tube from the laser 

which detects when the tridecane surface and the scale mark to be calibrated are in the same 

plane by indicating a sharp decrease in the intensity of the laser light. The laser power meter 

output (in Ω) is acquired by the computer and used along with the bath-and-tube vertical stage 

by the surface positioning process. A computer program controls the number of replications at 

each mark and then moves the entire process onto the next mark to be calibrated. The automated 

hydrometer calibrator is capable of applying Cuckow’s method at any number of scale marks on 

the hydrometer scale without operator intervention. It will operate unattended overnight. 
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Figure 3. Surface positioning process 

The apparatus can be operated in a semi-automatic mode wherein the operator centers the laser 

sheet on the scale mark to be calibrated by turning the lead screw by hand while looking directly 

at the scale or by looking at the CCD camera image of the scale on the computer monitor. Then 

the automatic features move the tridecane surface, determine when the tridecane surface is at the 

scale mark to be calibrated, and records the required data. The semi-automatic mode is faster 

than the fully automatic mode. 

One of NIST’s check standard hydrometers is calibrated immediately before each customer 

calibration. The tolerance of the check standard is 10–4 of the density. The check standard 

ensures that the entire calibration system (which includes the balance, the density of the 

tridecane, the surface tension of the tridecane, etc.) is operating within its expected range. 

5. Surface Positioning Process 

In the NIST apparatus, the laser light sheet is centered on the hydrometer scale mark to be 

calibrated and the tridecane surface is aligned with the center of the laser light sheet by lowering 

the tridecane and looking for the decrease in intensity of the laser light measured by the laser 

power meter. A plot of the laser light intensity versus elevation of the tridecane surface is shown 

in Figure 4. Figure 4 actually shows traces for three measurements at the same calibration mark 

to demonstrate the repeatability of the surface positioning process. The tridecane surface is 

raised above the mark to be calibrated by 3 mm and then lowered until the light intensity is at the 

half way point of the transition to ensure a stable and reproducible meniscus on the hydrometer 

stem. In Figure 4, at elevations from 51 mm to 54.8 mm the laser power meter outputs 

approximately 3000 Ω. At these elevations the laser sheet is passing through the walls of the tube 

and the bath, the water in the bath, and through air above the tridecane surface. At about 55 mm 

the light intensity moves through the transition and the power meter output rises from 3000 Ω to 

4500 Ω. At tridecane elevations of 55.4 mm and above, the power meter registers the highest 

intensity (4500 Ω). At these elevations the laser sheet is passing through the container walls, the 

water, and the tridecane. The three traces for the descending surface end at an elevation of about 

55.2 where the surface positioning process has determined that the light intensity is at the 

midpoint of the transition. At this point the tridecane surface is aligned with the scale mark to be 

calibrated and after a 40 s wait for stabilization of the balance, its readings along with the 

tridecane temperature are recorded to a computer data file for later processing. The vertical stage 

changing the tridecane surface elevation is operated at a slow speed (0.5 mm/min) so that the 
Hydrometer Calibration Service NIST SP 250-78

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surface does not over run the desired position due to computer processing or actuation delays. 

2500

3000

3500

4000

4500

5000

51 53 55 57

position (mm)

Power Meter Output (

Ω)

Figure 4. Laser power meter output versus the elevation of the liquid tridecane surface. As the 

liquid surface traverses the laser light sheet, the power meter registers the transition that we use 

in the surface positioning process. 

The shape of the light intensity transition is caused by 1) the lens behavior of the tridecane filled 

cylinder, 2) the vertical intensity profile of the laser light sheet, and 3) refraction and reflection 

of the light sheet by the tridecane meniscus. The large change in intensity is due to the vessel of 

tridecane acting as a cylindrical lens, focusing the light onto the power meter window. The width 

of the transition (about 0.2 mm) is due to the width of the laser light sheet. As the tridecane is 

raised and a larger portion of the light sheet passes through the liquid lens, the power meter 

readings increase. Therefore, at the midpoint of the transition, the middle of the laser light sheet 

is aligned with the bottom of the meniscus. Since we align the middle of the light sheet with the 

hydrometer mark to be calibrated, the bottom of the meniscus is aligned with the hydrometer 

mark to be calibrated. 

The decrease in intensity seen between 51 mm and 55 mm in Figure 4 is caused by refraction 

and reflection of the laser light by the tridecane meniscus inside the cylinder. The angle formed 

by the tridecane at the cylinder wall refracts or reflects (depending on the angle of the meniscus 

surface) the laser light away from the power meter window. The hysteresis in the power meter 

output between the rising versus descending tridecane surface is caused by the non-zero contact 

angle on the cylinder and hydrometer. 

6. Surface Tension and Contact Angle Effects 

Surface tension and contact angle effects are the primary reason for our selecting tridecane as the 

liquid of known density. The surface tension of tridecane is 25 mN/m at 20 °C. This is 

approximately 1/3 of the surface tension of water at 20 °C, consequently, surface tension and 

contact angle effects are approximately 1/3 of those for water. The surface tension of tridecane 

(and hydrocarbon liquids in general) has a weak sensitivity to contaminants. We found that the 

Tridecane surface 

transition 

Laser sheet 

through air 

Laser sheet 

through tridecane

Rising 

tridecane 

Descending 

tridecane 
Hydrometer Calibration Service NIST SP 250-78

 Page 12 October 20, 2011 

contact angle measured by the sessile drop method is effectively zero following a simple rinse 

with ethanol and drying with a clean paper towel. 

Initially we used distilled water as the liquid of known density in our apparatus. We had 

difficulty obtaining reliable results due to the sensitivity of water’s surface tension and contact 

angle to contaminants on the water’s surface or on the hydrometer stem. Water has large surface 

tension, 73 mN/m at 20 °C; however, trace contamination by detergents or hydrocarbons will 

reduce its surface tension by as much as 10 mN/m. An error in surface tension of 10 mN/m leads 

to errors of 150 parts in 106

 for 1 g/cm3 hydrometer. 

The contact angle formed by the water with the glass hydrometer stem is also sensitive to 

contamination that leads to unreliable results. The force due to surface tension on the glass 

hydrometer stem is πDγ cosθ where θ is the contact angle of the liquid with the hydrometer 

stem [12]. Derivations of the equations for Cuckow’s method generally assume that the contact 

angle is exactly zero, so that the surface tension force is πDγ . This assumption is made because 

it is particularly difficult to measure the contact angle against a curved surface such as a 

hydrometer stem and because the stem is assumed to be cleaned well enough that the contact 

angle is effectively zero. However, we have found that many glass cleaning procedures do not 

result in an effectively zero contact angle and that even if the stem is clean, it will not remain 

clean enough over a period of hours in our laboratory. In particular, glass cleaned with solvents, 

finished with ethanol, and air dried does not achieve an effectively zero contact angle. We note 

that if a hydrometer is cleaned with chromic-sulfuric acid, rinsed with water, air dried, and used 

immediately, the contact angle is less than 10°, which is close enough to zero. However, the 

stem will not remain clean for long unless carefully protected from contamination, such as oil 

droplets in air. 

7. Uncertainty of the NIST Hydrometer Calibration 

We analyzed the uncertainty of a density measurement caused by the NIST hydrometer 

calibration results [A and B in Equation (2)] following the method of propagation of uncertainties 

[13, 14, 15]. This involved obtaining normalized sensitivity coefficients, ( )( ) i L L i S = 1 ρ ∂ρ ∂x , 

by partial differentiation. We also obtained standard uncertainties ( i dx ) for the calibration inputs 

from experiments with the instruments in the apparatus. We examined a number of cases and 

found that the calibration results from our apparatus normally have an expanded uncertainty 

(coverage factor k = 2) of 100 parts in 106

 or less. In the following section, we consider 

uncertainty components that arise during application of the hydrometer (temperature effects, the 

customer’s surface tension and contact angle) and find that they can lead to uncertainties of 370 

parts in 106

 (k = 2), or larger. 

 

[12] Adamson A W and Gast A P 1997 Physical Chemistry of Surfaces, 6th ed., (New York, John Wiley 

and Sons) 

[13] International Organization for Standardization 1996 Guide to the Expression of Uncertainty in 

Measurement, (Switzerland, International Organization for Standardization) 

[14] Taylor B N and Kuyatt C E 1994 Guidelines for Evaluating and Expressing the Uncertainty of NIST 

Measurement Results, NIST Technical Note 1297, (Gaithersburg, National Institute of Standards and 

Technology) 

[15] Coleman H W and Steele W G 1999 Experimentation and Uncertainty Analysis for Engineers, 2nd ed 

(New York, John Wiley and Sons) 
Hydrometer Calibration Service NIST SP 250-78

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Table 1 gives an uncertainty analysis for a hydrometer that we used to measure the density of 

water. This hydrometer is similar in design to those that we receive for calibration. It has a span 

of 0.05 g/cm3

 and each scale increment is 1.3 mm long and represents 0.0005 g/cm3. The 

uncertainty analysis in Table 1 includes only the components related to the calibration at NIST 

using NIST apparatus. Table 1 has columns for: 1) the input quantities to Equation (1), 2) 

symbols xi for each input quantity, 3) their values, 4) the values of the standard uncertainties (k = 

1), 5) the units for each quantity, 6) the normalized sensitivity coefficients, 7) the values for the 

sensitivity coefficients, and 8) the product of the normalized sensitivity coefficient and the 

standard uncertainty, expressed as parts in 106

. The following numbered paragraphs correspond 

to the numbered k = 1 uncertainty components listed in Table 1 and explain their origin. 

1. We measured the density of the tridecane used in our apparatus by applying Arcimedes’ 

principle to a spherical stainless steel sinker of known mass and known volume at 

temperatures between 15 °C and 23 °C. The standard uncertainty of the Archimedes 

apparatus is 5 parts in 106

. The standard deviation of the density data from a linear function 

of the temperature was 2 parts in 106

. The uncertainty of the temperature of the tridecane in 

the hydrometer calibration apparatus is 0.008 K. This contributes to the tridecane density 

uncertainty through the thermal expansion coefficient βT = 710×10–6 K-1. Taking the RSS of 

the components listed above gives a standard uncertainty of the tridecane density of 7 ×10-6

g/cm3

 or 9 parts in 106. (We also checked the tridecane density using a vibrating tube 

densimeter that had been calibrated with air and distilled water. The densimeter results 

agreed with the Archimedes’ apparatus results within 37 parts in 106

, within the uncertainty 

of the densimeter.) 

2. Hydrometers are made from a variety of glasses, including borosilicate glass and soda-lime 

glass. The volumetric thermal expansion coefficient of these glasses varies from 9 × 10-6 K-1

to 26 × 10-6 K-1. Unless a reliable value is specified, this component has a relatively large 

standard uncertainty and contributes significantly to the expanded uncertainty of the NIST 

calibration. This component can be reduced to negligible levels by operating the apparatus at 

the hydrometer reference temperature. In this example we have operated the apparatus at 

20 °C and used a reference temperature of 15.56 °C (60 °F) to illustrate that the component is 

significant (22 parts in 106

 here). 

3. The temperature of the tridecane was spatially surveyed and monitored with a thermister. 

The tridecane temperature within the tube was uniform within 0.003 K and stable within 

0.005 K during the 3 hour interval necessary to calibrate a hydrometer. The results of three 

annual calibrations of this thermister are consistent within 0.005 K. The root-sum-square 

(RSS) of these temperature components is 0.008 K; we will use this value as the standard 

uncertainty of the temperature of the tridecane. 

5. and 6. Zero drift of the balance is the primary source of uncertainty for the apparent masss 

measurement (0.3 mg). Zero readings are checked before and after hydrometer calibrations 

and zero readings in error by more than 3 mg trigger re-testing. Repeatability, resolution (0.1 

mg), linearity (checked periodically with calibration masses), and the uncertainty of 

calibration masses are negligible in comparison. For the dry hydrometer measurement, the 

balance’s resolution and linearity are the most significant components and the standard 

uncertainty is 0.15 mg. 
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Table 1. Uncertainty analysis for a hydrometer calibrated with the NIST Cuckow’s apparatus. 

See Table 2 for extra components that arise during hydrometer usage. 

Quantity i x i x

Value 

i dx

Value 

units Si

i S 

Value 

106 Si 

dxi

1) Density of 

tridecane ρ T 0.756 7×10-6 g/cm3

(O O m V )

O O m

T 2 3 s T s

2 3 s

ρ − + − ρ

− +

1.32 9 

2) Hydrometer 

volume thermal 

expansion 

β 2.5×10-5 5×10-6 1/K TT – T0 4.44 22 

3) Temperature of 

tridecane 

TT 20.0 0.008 °C 2 3 s s T

2 3 s T s s s T

ρ

β ρ β ρ

O O m V

(O O m V ) V

− + −

− + − +

2.5×10-5 0 

4) Density of air ρ a2 1.2×10-3 1×10-5 g/cm3

T 2

3 s T s

O

O m V

ρ

− + − ρ -0.32 -3 

5) Hydrometer 

apparent mass 

in air 

O2 48 0.00015 g O (O O m V )

O m V

2 2 3 s T s

3 s T s

ρ

ρ

− + −

− + − -6.6×10-3 -1 

6) Hydrometer 

apparent mass 

in tridecane 

O3 11.6 0.0003 g 

2 3 s T s

1

O − O + m − ρ V 2.8×10-2 8 

7) Stem diameter D 0.5 0.001 cm ( ) ( ) ⎥

⎦

⎤ ⎢

⎣

⎡

− + − −

2 3 s T s

T T

2

L L cos cos

g O O O m ρ V

π γ θ γ θ -2.3×10-3 2 

8) Surface tension 

of tridecane Tγ 25 0.5 mN/m ( )

g(O O m V )

D

s ρ T s

π θ

− + − −

2 3

T cos

-4.4×10-5 -22 

9) Contact angle 

in tridecane 

( ) T cos θ

1 0.015 

g(O O m V )

D

s ρ T s

π γ

− + − −

2 3

T -1.1×10-3 -17 

10) Acceleration 

due to gravity g 980 1×10-2 cm/s2 ( ) ( ) ⎥

⎦

⎤ ⎢

⎣

⎡

− + − −

2 3 s T s

T T

2

L L

2

cos cos

g O O O m V

D

ρ

π γ θ γ θ -1.2×10-6 0 

11) Mass of sinker ms 0 0∗

 g 2 3 s T s

1

O − O + m − ρ V

− -2.8×10-2 0 

12) Volume of 

sinker 

Vs 0 0*

 cm3

2 3 s T s

1

O −O + m − ρ V 2.8×10-2 0 

13) Thermal 

expansion of 

sinker volume 

β s 0 0*

 1/K 

2 3 s T s

T s T 0

O O m V

V (T T )

ρ

ρ

− + −

− 0.74 0 

14) Laser 

positioning 8.3×10–3 cm ∂ l

∂ L

L

1 ρ

ρ

3.7×10-3 31 

15) Surface 

positioning 2×10-3 cm ∂ l

∂ L

L

1 ρ

ρ

3.7×10-3 7 

16) Repeatability 1 1×10-5 g/cm3 1 10 

Expanded Uncertainty of NIST Calibration RSS x 2 101 

 

∗

 Sinker not used for the hydrometer used in this example. 
Hydrometer Calibration Service NIST SP 250-78

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8. We measured the surface tension of our tridecane with a DuNouy tensiometer and our 

measurements differed from literature values [16] by 1 mN/m or less. We have taken 

0.5 mN/m as the standard uncertainty. 

9. Our experiments showed that the contact angle for tridecane-air interface on glass is small 

after the glass is rinsed in ethanol and dried with a paper towel. Our calculations assume the 

contact angle is zero, but we will use a standard uncertainty of 10° for our uncertainty 

analysis. It is convenient to express the uncertainty in terms of ( ) T cos θ and ( ) L cos θ rather 

thanθ T andθ L . 

10. The acceleration of gravity g at the apparatus is 980.1018 cm/s2

 with a standard uncertainty 

of 10 parts in 106

 or less. [17] 

11. , 12., and 13. The sinker is needed only when calibrating hydrometers for liquids less dense 

than tridecane (<0.76 g/cm3

). When the sinker is used, the sinker uncertainty components are 

all less than 6 parts in 106

, which is negligible compared to other components. 

14. The largest uncertainty component is one that is not explicitly found in Equation (1): the 

alignment of the laser light sheet with the hydrometer stem mark. The uncertainty for this 

component depends on the pixel resolution of the camera system that we use during 

alignment. The standard uncertainty is one half of the pixel dimension (0.016/2 = 0.08 mm). 

The normalized sensitivity coefficient for this component is ( )(∂ ∂l) 1 ρ L ρ L where l is the 

length along the hydrometer stem. For a hydrometer of the design used in this example 

(1.3 mm scale increments represent 0.0005 g/cm3

), 0.08 mm length uncertainty corresponds 

to 31 parts in 106

 density uncertainty. This uncertainty value is not changed when the laser is 

aligned with the scale mark by the operator instead of by the computer. We have calibrated 

the same stem mark repeatedly, manually moving the laser light sheet off and back onto the 

mark each time, and we have found the standard deviation of the calibration results to be 20 

parts in 106

. 

15. We also include a component due to alignment of the tridecane surface with the laser light 

sheet by the surface positioning process. The process reliably locates the midpoint of the 

power meter transition within 10 % of the transition. The slope of the transition (see Figure 

4) allows us to translate the 10 % uncertainty into a length and then a density uncertainty for 

the hydrometer stem design (7 parts in 106

). This value is confirmed by repeatability 

measurements: when we leave the laser light sheet in a fixed position and use the surface 

positioning process repeatedly, the liquid surface position (as measured by the linear 

encoder) has a standard deviation of 0.016 mm, 6 parts in 106

 for density. 

16. All of the uncertainty components discussed up to now are of type B. For a customer 

calibration, we include the standard deviation of repeated measurements made at the same 

scale mark in the uncertainty analysis (type A). In our experience, this is always less than 10 

parts in 106

 and we have included a component of this size in Table 1. 

Taking the root-sum-square of the components and multiplying by 2 gives an expanded, 

approximately 95 % confidence level uncertainty. The result is 101 parts in 106

 (k = 2) for a 

hydrometer of the design type we calibrate most often. 

 

[16] Jasper J J, Kerr E R, and Gregorich F 1953 The Orthobaric Surface Tensions and Thermodynamic 

Properties of the Liquid Surfaces of the n-Alkanes, C5 to C 28, J Am Chem Soc, 75, 5252-5254 

[17] Tate D R 1967 Acceleration Due to Gravity at the National Bureau of Standards, J. of Res. of the 

National Bureau of Standards-C, Engineering and Instrumentation, 72C, No. 1 

 
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In this example, the laser positioning component is the largest contributor. This component is 

analogous to the ability to “read” the position of the liquid surface on the hydrometer scale and 

we can do this with our apparatus with standard uncertainty of 0.08 mm. This length corresponds 

to different density uncertainties depending on the design of the hydrometer under test. For a 

high resolution hydrometer, this component can drop to less than half the value in Table 1. Then, 

the dominant uncertainty components of our apparatus become surface tension, contact angle, 

and the hydrometer thermal expansion coefficient, and the expanded uncertainty of the 

calibration result falls below 100 parts in 106

 (k = 2). 

8. Uncertainty When the Hydrometer Is Used 

Table 2 shows uncertainty analyses for hydrometers used to measure the density of water and 

two hydrocarbon liquids (trichloroethylene and toluene). The table gives sample magnitudes for 

illustration, but users’ values may be quite different. The examples are given to show hydrometer 

users that uncertainties other than the NIST calibration must be considered and that those extra 

uncertainties are significant. 

Surface tension and contact angle are needed when Equations (1) or (2) are used to calculate the 

density of a liquid. The uncertainty of these values is dependent on the liquid under test. For 

hydrocarbon liquids, handbook surface tension values are available and uncertainties of 1 mN/m 

are reasonable. Simple cleaning procedures will give contact angles of nearly zero with standard 

uncertainty of 10° for hydrocarbons on glass. For water and aqueous solutions, only certain 

cleaning procedures will give contact angles less than 20° (for instance chromic-sulfuric acid, 

water rinsed, air dried) and unless these procedures are used, large uncertainties must be 

assumed. This problem is compounded by the large values of surface tension for aqueous 

solutions and their large uncertainties since they are sensitive to contamination. 

Uncertainty due to thermometer calibration and temperature non-uniformity of the liquid sample 

in which the hydrometer is floated can be large. The sensitivity coefficient is proportional to the 

volumetric thermal expansion coefficient of the liquid under test, about 200 × 10-6 K-1 for 

aqueous solutions, about 1000 × 10-6 K-1 for most hydrocarbons, and 1670 × 10-6 K-1 for 

trichloroethylene. The temperature uncertainty used in Table 2 is half the measurement 

resolution recommended in an ISO hydrometer procedure and one tenth of the allowed 

difference between before and after temperature measurements [18]. 

 

[18] International Organization for Standardization 1981 ISO 649/2 Laboratory Glassware – Density 

Hydrometers for General Purposes – Part 2: Test Methods and Use, (Switzerland, International 

Organization for Standardization) 
Hydrometer Calibration Service NIST SP 250-78

 Page 17 

Table 2. Uncertainty for hydrometer density measurements in water, trichloroethylene, and toluene. 

Liquid under test Water Trichloroethylene Quantity 

xi units 

Si xi dxi Si

106 ×

Si dxi xi dxi Si

106 ×

Si dxi 

NIST calibration 

A, B 1 50 1 44 

User surface tension 

L

γ

mN/

m 

( )

2

L cos

gO

πD

θ

68 2.5 3.1×10-5 78 29 1 3.1×10-5 17 

User contact angle 

( ) L cos

θ

2

L

gO

πDγ 0.94 0.045 2.3×10-3 -102 1 0.015 5×10-4 8 

T effects 

TL °C 

∂ l

∂

L

L

1 ρ

ρ

20 0.1 2×10-4 20 20 0.1 1.7×10-3 167 

Resolution 

ρ L g/cm3 1.0 1 × 10-4 100 1.46 1 × 10-4 68 

Expanded Uncertainty of Hydrometer Usage RSS 

× 2 344 RSS 

× 2 373 
Hydrometer Calibration Service NIST SP 250-78

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Two additional uncertainty components deserve mention: the density of air and the gravitational 

constant. Cuckow’s derivation assumes that the air density buoying the emergent stem is equal 

during calibration and usage. We examined the case where a hydrometer was calibrated at sea 

level and used at an elevation of 2000 m and found that Cuckow’s approximation introduced an 

error of less than 10 parts in 106

, negligible relative to the uncertainties in Table 2. We also note 

that the two instances of g found in Equation (1) may differ since calibration and usage may 

occur at different latitudes or altitudes. A difference in g of 0.25 % leads to an error of less than 

10 parts in 106

. 

For water, the dominant uncertainty contributors are the surface tension and contact angle. For 

trichloroethylene, temperature effects dominate all others due to a large thermal expansion 

coefficient. In the example for a hydrometer used in toluene, the hydrometer scale has twice the 

resolution of the prior examples. This allows lower uncertainty from our apparatus (77 parts in 

106

, k = 2) and during usage (238 parts in 106, k = 2). 

9. Validation with Liquids of Known Density 

The performance of the apparatus was validated with a two step process. First we freely floated a 

hydrometer in a liquid of known density. We stirred the liquid, measured its temperature, floated 

the hydrometer and read it by eye, removed the hydrometer, stirred the liquid, and measured the 

temperature of the liquid again. We then used our apparatus and Equation (1) to calculate the 

density represented by the same stem scale mark read on the freely floating hydrometer. We 

performed the validation tests in four liquids; tridecane, toluene, water, and trichloroethylene. 

The results of the validation tests are listed in Table 3. The densities based on temperature 

measurements and from the Cuckow’s apparatus all agree within 0.00003 g/cm3

 (40 parts in 

106

). 

Table 3. Summary of validation tests using liquids of known density. 

Liquid Density from 

equation of state 

(g/cm3

) 

Density from 

Cuckow’s 

apparatus 

(g/cm3

) 

Difference 

(g/cm3

) 

Difference 

(parts in 106) 

Tridecane 0.75695 0.75692 -0.00003 -40 

Toluene 0.86779 0.86778 -0.00001 -12 

Water 0.99825 0.99824 -0.00001 -10 

Trichloroethylene 1.46550 1.46551 +0.00001 +6 

The density as a function of temperature (at atmospheric pressure) for tridecane and 

trichloroethylene were determined by Archimedes’ principle as described earlier (standard 

uncertainty 5 parts in 106

). (The same tridecane was used for the freely floating measurements 

and the Cuckow’s apparatus measurements). The density as a function of temperature for the 

toluene sample was measured with a vibrating tube densimeter (standard uncertainty 15 parts in 

106

). The density of the water was calculated using the equation of Patterson and Morris [1]. We 

 

[1] Patterson J B and Morris E C 1994 Measurement of Absolute Water Density, 1°C to 40°C, Metrologia, 
Hydrometer Calibration Service NIST SP 250-78

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compared the Archimedes’ apparatus and vibrating tube densimeter measurements of the density 

of our distilled water to the equation of Patterson and Morris and found agreement within 10 

parts in 106

. Therefore the standard uncertainty of the density as a function of temperature 

equations for the liquids used in the validation experiments is always less than 15 parts in 106

. 

The validation experiments apply hydrometers in much the same way that our calibration 

customers do. As a result, the validation experiments are subject to the same uncertainty 

concerns for the freely floating test that are listed in Tables 2 and 3, i.e., the Cuckow’s apparatus 

results, surface tension, contact angle, temperature, and resolution, as well as the density as a 

function of temperature uncertainties given above. We used chromic sulfuric acid to clean the 

hydrometer stems for the freely floating experiments and this is critical to obtaining low 

uncertainty for the water validation experiment. The density agreement found in the validation 

experiments (40 parts in 106

) is well within our expectations based on an uncertainty analysis 

(373 parts in 106

) for the experiment. 

10. Comparison to NIST Historical Results for a Reference Hydrometer 

NIST has a collection of reference hydrometers that have been used to calibrate customers’ 

hydrometers by the comparison method. The reference hydrometers have been calibrated by 

placing them in liquids of known density, said density determined by measuring the apparent 

mass of sinkers of known mass and volume (Archimedes’ principle). We have records of the 

calibration of one of these reference hydrometers (KNEK 260) by Bowman and Gallagher [2] 

(measured in 1956, published in 1969). We calibrated KNEK 260 in our Cuckow’s apparatus at 

three different tridecane temperatures and the results, along with those of Bowman and 

Gallagher are shown in Figure 5. The old and new calibrations agree within 160 parts in 106

 over 

the range of density tested. 

 

The three sets of data from our apparatus differ by as much as 40 parts in 106

, and appear to be a 

function of test temperature. We believe the apparent temperature dependence is due to a small 

error in the value β = 23 × 10-6 / K. that Bowman and Gallagher used for the volumetric thermal 

expansion coefficient of the KNEK 260 hydrometer. If we replace the Bowman and Gallagher 

value of β with the value β = 19 × 10-6 / K, the temperature dependence vanishes. 

 

31, 277-88 

[2] Gallagher W H and Bowman H A 1969 An Improved High-Precision Calibration Procedure for 

Reference Standard Hydrometers, J. of Res. of the National Bureau of Standards-C, Engineering and 

Instrumentation, 73C, Nos. 3 and 4 
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0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

0.0011

0.0012

0.95 0.97 0.99 1.01 1.03

Specific Gravity

Correction (-)

Bowman & Gallagher, 1969

This work 25 °C

This work 19 °C

This work 16 °C

Figure 5. Corrections for hydrometer KNEK 260 from our apparatus (used at three tridecane 

temperatures) and from Bowman and Gallagher, 1969. Uncertainty bars are 100 parts in 106

from Table 1. 

11. Summary 

We constructed and validated an automated standard that calibrates hydrometers by Cuckow’s 

method. The advantages of the automated hydrometer calibrator include: 

1. It can calibrate any number of hydrometer scale marks with any number of replications 

while unattended by an operator. Thus a hydrometer can be calibrated overnight. 

2. Any location on the hydrometer scale can be calibrated. This is superior to the 

comparison method of calibration where the location on the scale of the calibration is 

dictated by the density of the liquid used for the calibration. 

3. Only one calibrating fluid is used and it is user-friendly. In the comparison method, 

several mixtures of sulfuric acid/water and volatile organic solvent/oil mixtures are 

required. 

Calibration results from the new standard will be presented in the format of Equation (10). To 

obtain the best results, customers must know the surface tension of the test liquid. Ignoring the 

surface tension term completely (i.e., assuming cos 0 Bγ L θ L = ) will lead to errors of about 

0.002 g/cm3 for aqueous solutions and about 0.001 g/cm3 for hydrocarbon liquids. Handbook 

values are available for many liquids and the NIST Fluid Metrology Group can assist customers 

that need help determining the surface tension of the liquid they are working with. 

We selected tridecane as the liquid of known density in our Cuckow’s apparatus since it has 

relatively low surface tension (25 N/m) that is not sensitive to contamination, and it forms a 

contact angle less than 10° with the glass hydrometer stem even with a simple ethanol cleaning 

process. It has a low vapor pressure, so we do not lose liquid due to evaporation at an 

unacceptable rate. It is a single component fluid of high purity: the liquid that remains will 

maintain a constant density as some of it evaporates. 
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The uncertainty of the calibration at NIST is 100 parts in 106

 for a typical hydrometer design. 

Usage of a hydrometer to measure an unknown liquid density introduces several significant 

uncertainty components (surface tension, contact angle, temperature effects, and resolution) that 

lead to uncertainties of 370 parts in 106

 or more unless great care is taken. This is particularly 

true for aqueous solutions or liquids with large thermal expansion coefficients. 

We validated our automated Cuckow’s apparatus using four liquids of known density and found 

agreement of 40 parts in 106

 or less, well within our uncertainty specifications. We compared the 

results of our apparatus with historic calibrations of one of our reference hydrometers (KNEK 

260). KNEK 260 is one of the hydrometers we use as a working standard to perform comparison 

method calibrations of customer hydrometers. We found that the old and new calibrations agree 

within 160 parts in 106

, indicating that changing to our new calibration method should not 

introduce any significant discontinuity in the results of calibrations performed by NIST. We 

tested KNEK 260 at tridecane temperatures ranging from 16 °C to 25 °C and obtained consistent 

results within our uncertainty for the thermal expansion coefficient of the hydrometer. 
Hydrometer Calibration Service NIST SP 250-78

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Appendix A. Corrections to Hydrometer Readings for Surface Tension Effects 

A hydrometer floating in a liquid is subject to the forces shown in Figure A1: liquid buoyancy, 

air buoyancy, gravity, and surface tension effects. 

Figure A1. Schematic of a freely floating hydrometer with representations of the forces imposed on 

it. 

Hence, while primarily a function of liquid density ρ L , hydrometer readings are also dependent 

on the liquid surface tension L γ and the contact angle between the liquid surface and the 

hydrometer stem θ L . Most liquids partially or completely wet clean glass. For such liquids, a 

curved liquid-vapor interface (meniscus) rises up the hydrometer stem and, in effect, pulls the 

hydrometer deeper into the liquid. 

Expressed in units of density, the effect of surface tension on the reading of a hydrometer is: 

g m

D

g V

D

c L L L mid π γ cosθ π γ ρ

γ = ≅ , (A1) 

where D is the hydrometer stem diameter, V is the volume of the submerged portion of the 

hydrometer, g is the gravitational constant, L γ is the surface tension of the liquid being 

measured, andθ L is the contact angle of the liquid on the hydrometer stem (normally assumed to 

be zero). For most applications, the second, approximate expression in Equation 1 suffices: the 

contact angle is assumed to be zero and the submerged volume is approximated by the mass of 

the hydrometer m divided by the density represented by the mid-range of the hydrometer 
Hydrometer Calibration Service NIST SP 250-78

 Page 23 October 20, 2011 

scale, ρ mid . 

When aqueous solutions contact a clean, glass hydrometer, the rising meniscus can alter the 

hydrometer’s equilibrium position by the equivalent of up to 0.2 % reduction of the liquid 

density. For hydrocarbon liquids, the reduction of the apparent density can be as large as 0.1%. 

To determine whether on not surface tension corrections are significant for a particular 

hydrometer application, we can use Equation A1 to calculate the effect of a 1 dyne/cm surface 

tension change. For a typical hydrometer (submerged volume 50 cm3

, stem diameter 0.5 cm), a 1 

dyne/cm surface tension change causes a reading change of 3.2 × 10-5 g/cm3

. Aqueous solutions 

are particularly susceptible to surface tension changes due to contaminants. Distilled water has a 

surface tension of approximately 73 mN/m while distilled water contaminated with oil has a 

surface tension of approximately 60 mN/m. Hence a hydrometer that was calibrated for distilled 

water will read high by 0.0004 g/cm3

 in water contaminated with oil. Distilled water 

contaminated with soap will have a surface tension of approximately 30 mN/m and a hydrometer 

calibrated for the surface tension of distilled water will read high by 0.0013 g/cm3

. These 

calculations assume that the contact angle is zero during hydrometer calibration and usage. If the 

contact angle is zero during calibration (well cleaned) and non-zero during usage 

(contaminated), the density errors will be worse than the estimates given above. 

In the following case studies, the liquid density ρ is the hydrometer reading R, plus a 

hydrometer correction C: 

s γ ρ = R + C = R + c + c . (A2) 

As shown in Equation A2, the hydrometer correction can be broken into two parts. The quantity 

s c is the scale correction, an offset due to vertical misplacement of the hydrometer scale within 

the stem. It would be the correction to the hydrometer for a liquid with zero surface tension. The 

quantity γ c is the portion of the hydrometer correction that can be attributed to surface tension 

effects as given in Equation A1. 

The effects of surface tension for the following four cases will be covered: 

1) hydrometer calibrated by hydrostatic weighing and used to measure a liquid density, 

2) hydrometer calibrated in a liquid of known density and used to measure the density of a liquid 

with a different surface tension, 

3) calibrations by the comparison method, and 

4) hydrometer calibrated by the comparison method and used to measure the density of a liquid 

with a different surface tension. 

Hydrometer Calibrated by Hydrostatic Weighing and Used to Measure a Liquid Density

Calibration results from hydrostatic weighing (Cuckow’s method3

) are given as a correction to 

the hydrometer reading in the following form: 

 

3 Cuckow F W 1949 A New Method of High Accuracy for the Calibration of Reference Standard 

Hydrometers, J. Soc. Chem. Industry, 68, 44-9
Hydrometer Calibration Service NIST SP 250-78

 Page 24 October 20, 2011 

L s L L C = c + Bγ cosθ (A3) 

where s c and B are the calibration outputs, L γ is the surface tension (in mN/m) and θ L is the 

contact angle for the liquid under test. s c has the same units as the hydrometer scale and B has 

the units of the hydrometer scale units divided by mN/m. 

Example A1: 

Given: A hydrometer is used in a liquid with surface tension of 26 mN/m and the reading is 

1.025 g/cm3

. The values of s c and B for the 1.025 position on the hydrometer are 0.0010 g/cm3

and 3 × 10-5 [(g/cm3

)/(mN/m)]. What is the liquid density according to the hydrometer? Assume 

the hydrometer is well cleaned and that the contact angle is zero. 

Solution: Using Equation A3, the hydrometer correction is: 

cos 0.0010 (g/cm ) 3 10 [(g/cm /(dynes/cm)] 26 (dynes/cm) cos (0 ) 3 5 3

L s L L

− o C = c + Bγ θ = + ×

( ) 3

L C = 0.0018 g/cm , and using Equation A2 (C = CL ), the liquid density is: 

( ) ( ) ( ) 3 3 3 ρ = R + CL = 1.025 g/cm + 0.0018 g/cm = 1.0268 g/cm

Hydrometer Calibrated in a Liquid of Known Density and Used to Measure the Density of a 

Liquid with a Different Surface Tension 

A hydrometer can be calibrated by floating it in a reference liquid of known density, reading the 

hydrometer, and calculating the hydrometer correction using a rearranged version of Equation 

A2. One method of determining the reference liquid density is by Archimedes’ principle using a 

sinker of known volume and mass. Alternatively, one could use literature density values for high 

purity liquids. 

Surface tension affects the hydrometer reading during the calibration, but if the hydrometer is 

subsequently used to measure the density of liquid with the same surface tension, the hydrometer 

correction remains valid. However, if the hydrometer is used to measure the density of a liquid 

with a different surface tension, the change in a density measurement is: 

( ) γ L R γ γ

π Δ = − g V

D

c (A4) 

where L γ is the surface tension of the liquid being measured, and R γ is the surface tension of the 

reference liquid used to calibrate the hydrometer. A contact angle of zero for both liquids is 

assumed in Equation A4. As given in Equation A1, a reasonable approximation to the submerged 

volume is the hydrometer mass divided by the liquid density represented by the mid-range of the 

hydrometer scale. 

Once the change in surface tension correction γ Δc has been calculated, it is added to the 

hydrometer reading and the correction determined in the reference liquid to obtain the density of 
Hydrometer Calibration Service NIST SP 250-78

 Page 25 October 20, 2011 

the liquid under test: 

( ) ( ) R γ s γR γL γR s γL R CL ρ = R + C + Δc = R + c + c + c − c = R + c + c = + (A5) 

Note that Equations A4 and A5 are in density units and for hydrometers with any units other 

than density (i.e. specific gravity, percent alcohol, API, Baumé), hydrometer readings and the 

change in surface tension correction must be put into consistent units before they can be added 

(see Example A2). 

Example A2:

Given: A specific gravity 60/60 °F hydrometer with a range of 0.95 to 1.0 was calibrated in 

distilled water and the hydrometer correction was found to be +0.0010 at a reading of 0.9995. 

The hydrometer is subsequently used to measure the density of an oil and the hydrometer reading 

is also 0.9995. The surface tensions of the water and of the oil are 75 mN/m and 25 mN/m, 

respectively. The hydrometer weighs 50 g and its stem diameter is 0.5 cm. What is the specific 

gravity of the oil? 

Solution: Calculate an approximate submerged volume for the hydrometer using its mass, the 

mid-range of the hydrometer stem, and the density of water at 60 °F: 

( ) ( ) ( )

3

3

mid mid 2

51.33 cm

0.975 0.999016 g/cm

50 g

sp gr 60 60 60 F = = = = o

H O

m m V

ρ ρ

Use Equation A4 to determine the change in hydrometer reading (in density units) caused by the 

surface tension difference. Note that 1 dyne/cm ≡ 1 g/s2

: 

( ) ( )

( )( ) 2 2 3

γ L R 3 2 25 g s 75g s 0.0016 g cm

51.33 cm 980.665cm s

0.5 cm Δ = − = − = − π

γ γ

π

g V

D

c

Apply the hydrometer correction determined with the reference liquid CR to the hydrometer 

reading R and calculate the density represented by the mark, absent surface tension corrections: 

( )( ) ( ) 3 3

R C H2O R + = 60 F R + C = 0.999016 g cm 0.9995 + 0.0010 = 0.9995 g cm R

o ρ ρ

Calculate the corrected oil density using the surface tension correction: 

( ) 3 3 3

R C γ 0.9995 g cm 0.0016 g cm 0.9979 g cm R ρ = ρ + + Δc = + − =

Calculate the specific gravity of the oil: 

( ) 0.9989

0.999016 g cm

0.9979 g cm

60 F sp gr 60/60 F 3

3

H2O

= = = o

o

ρ

ρ

Calibrations by the Comparison Method
Hydrometer Calibration Service NIST SP 250-78

 Page 26 October 20, 2011 

Hydrometers are often calibrated by comparing them to reference hydrometers floating in the 

same liquid. If the hydrometers have the same stem diameter and are well cleaned (i.e. same 

contact angle), the surface tension effects are the same for both hydrometers and surface tension 

corrections are unnecessary. However, it is important to note that it may be necessary to apply a 

surface tension correction to the reference hydrometer. For instance, if the correction for the 

reference hydrometer was determined using a liquid of one surface tension and then it was used 

in the comparison method in a liquid with a different surface tension, a procedure like that of 

Example A2 would be necessary to obtain correct measurements from the reference hydrometer. 

Similarly, the hydrometer correction determined by the comparison method for the hydrometer 

under test is valid if it is subsequently used in a liquid with the same surface tension, but if 

different liquids are used during calibration and subsequent usage, surface tension corrections 

based on Equation A4 are necessary (see Example A4). Hence, the surface tension of the liquids 

used during calibrations performed by the comparison method should be specified so that surface 

tension corrections can be made when the hydrometer under test is used later. 

Hydrometer Calibrated by the Comparison Method and Used to Measure the Density of a Liquid 

with a Different Surface Tension

In this circumstance it is necessary to know the surface tension of the liquid used to determine 

the hydrometer correction, the surface tension of the liquid that is being tested, the hydrometer 

mass, and the stem diameter. Equation A4 gives the effect of surface tension on the hydrometer 

reading in units of density. As for Example A2, if the hydrometer units are something other than 

density units (i.e. specific gravity, percent alcohol, API gravity), hydrometer readings and the 

change in surface tension correction must be put into consistent units before they can be added. 

Example A3:

Given: A 1.0000 to 1.0500 sp gr 60/60 °F hydrometer was calibrated by the comparison method 

and at a reading of 1.0350 its correction is -0.0008. The surface tension of the liquid used during 

calibration was 68 mN/m. The hydrometer mass is 50 g and its stem diameter is 0.5 cm. The 

hydrometer is being used to measure the density of a liquid with a surface tension of 33 mN/m 

and the reading of the hydrometer in the liquid under test is 1.0350. What is the specific gravity 

of the liquid under test? 

Solution: Using the same approach shown in Example A2, the approximate submerged volume 

of the hydrometer is: 

( )( ) ( )

3

3

mid mid H2O

48.83 cm

1.025 0.999016 g/cm

50 g

sp gr 60 60 F 60 F = = = = o o ρ ρ

m m V

Using Equation A4, the change in surface tension correction is: 

( ) ( )

( )( ) 2 2 3

γ L R 3 2 33 g s 68g s 0.0011 g cm

48.83 cm 980.665cm s

0.5 cm Δ = − = − = − π

γ γ

π

g V

D

c

Apply the hydrometer correction (CR ) to the hydrometer reading (R ) measured in the liquid 

under test and calculate the density represented by the mark, absent surface tension corrections: 
Hydrometer Calibration Service NIST SP 250-78

 Page 27 October 20, 2011 

( )( ) ( ) 3 3

R C H2O R 60 F 0.999016 g cm 1.0350 0.0008 1.0332 g cm R + = R + C = − = o ρ ρ

Calculate the corrected density for the liquid under test using the surface tension correction: 

( ) 3 3 3

R C γ 1.0332 g cm 0.0011 g cm 1.0321 g cm R ρ = ρ + + Δc = + − =

Calculate the specific gravity of the liquid under test: 

( ) 1.0331

0.999016 g cm

1.0321 g cm

60 F sp gr 60/60 F 3

3

H2O

= = = o

o

ρ

ρ

Appendix B: Corrections to Hydrometers for Temperature Effects 

Liquid temperature can affect hydrometer density measurements in two ways: 1) thermal 

expansion for the hydrometer material (generally glass) and 2) thermal expansion of the liquid 

under test. Hydrometers that read in density units need only be corrected for the volumetric 

thermal expansion of the hydrometer glass ( β ), nominally 26 parts in 106

 for each 1 °C 

temperature change. But consider a specific gravity hydrometer labeled with the sp gr 60 / 60 °F 

unit. This hydrometer is designed to give the density of the liquid under test at 60 °F relative to 

the density of water at 60 °F. Instructions for using such a hydrometer call for the liquid and the 

hydrometer to be uniformly at 60 °F during the measurement. If the liquid under test is not 

actually at 60 °F, corrections for thermal expansion of the hydrometer glass and of the liquid are 

necessary to obtain specific gravity at the reference temperature conditions. This is also true for 

API and Baumé hydrometers. The volumetric thermal expansion coefficient for the liquid must 

be known for the particular liquid under test. Volumetric expansion coefficients for common 

liquids are: water and aqueous solutions ≈ 200 ×10-6 / °C, oils ≈ 800 ×10-6 / °C, so temperature 

corrections due to the liquid are normally much more significant than those for the glass. 

Temperature Corrections for Density Hydrometers 

The corrections to be applied to a hydrometer, when used at temperatures other than its standard 

temperature, may be obtained as follows. Assume R to be the reading of a density hydrometer 

indicating density correctly when floating in a liquid at temperatureT0 , the standard temperature 

for the instrument. Then 

m = R V (B1) 

where m is the mass of the hydrometer and V is the submerged volume of the hydrometer. 

Now assume the hydrometer to be again reading R, while floating in a liquid at temperature T ′ . 

Let cT′ be the correction which must be added to R in order to give the density of the second 

liquid in g/cm3

 at T ′ . In this case, we have: 

 

( ) [1 ( ' )] T V g T T0 m = R + c ′ + β − (B2) 
Hydrometer Calibration Service NIST SP 250-78

 Page 28 October 20, 2011 

where β g is the volumetric thermal expansion coefficient of the glass of which the hydrometer is 

made. 

Solving Equations B1 and B2 for cT′ , we obtain: 

( )

( ) g 0

g 0

T 1 '

'

T T

R T T

c

+ −

− ′ = − β

β . (B3) 

Since β g is small compared to 1, we may write: 

( ) T g 0 c ′ ≅ −R β T'−T (B4) 

Example B1:

Given: A density hydrometer with reference temperature of 20 °C is used to determine the 

density of a liquid at 30 °C. The hydrometer reads 1.0000 g/cm3

 and the volumetric thermal 

expansion coefficient for the hydrometer glass is 26 ×10-6 / °C. What is the density of the liquid? 

Solution: Calculate the correction using Equation B4: 

( ) ( )( ) 3 6 3

T g 0 = − '− = −1 g/cm 26×10 / C 30 C − 20 C = −0.0003 g/cm −

′

o o o c Rβ T T

Add the correction to the reading to obtain the liquid density: 

3 3 3 ρ = R + cT′ = 1.0000 g/cm − 0.0003 g/cm = 0.9997 g/cm

Note that this is the density of the liquid at 30 °C, not its density at 20 °C. 

Temperature Corrections for Specific Gravity Hydrometers 

The temperature correction for specific gravity hydrometers is obtained as follows. Assume S to 

be the reading of a hydrometer indicating specific gravity correctly at T0 °C relative to the 

density of water at T0 °C (a sp gr T0 / T0 hydrometer). Then 

() () m = V ρ L T0 = V S ρ H2O T0 (B5) 

where m is the mass of the hydrometer, V is the submerged volume of the hydrometer, ( ) ρ L T0

is the density of the liquid under test at T0, and ( ) ρ H2O T0 is the density of water at T0. 

Now assume the hydrometer is placed in a liquid at T′such that the instrument again reads S. 

Let cT′ be the correction added to S to give the correct sp gr T0 / T0 for the liquid, then: 

( ) () [ ] ( )[ ( )] T g 0 H2O 0 L 0 m = S + c ′ V 1+ β T'−T ρ T 1+ β T'−T , (B6) 
Hydrometer Calibration Service NIST SP 250-78

 Page 29 October 20, 2011 

where β L is the volumetric thermal expansion coefficient of the liquid under test. In Equation 

B6, we have used the definition of sp gr 60/60 °F for the corrected hydrometer 

reading:( ) () () T L 0 H2O 0 S + c ′ = ρ T / ρ T . 

 Solving Equations B5 and B6 for cT′ , we obtain: 

[ ] ( )[ ] ( )

[ ] ( )[ ] ( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ − + −

− + − + − ′ =

L 0 g 0

L 0 g 0

T 1 ' 1 '

1 1 ' 1 '

T T T T

T T T T

c S

β β

β β , (B7) 

from which it is clear that the expansion of the liquid as well as the expansion of the hydrometer 

glass must be taken into account. Neglecting second order terms, Equation B7 can be simplified 

to: 

( )( ) T 0 L g c ′ ≅ −S T'−T β + β (B8) 

Example B2:

Given: A specific gravity 60/60 hydrometer is used to measure the density of an oil sample that 

is at 25 °C and it reads 0.7800. The volumetric thermal expansion coefficient of the oil is known 

to be 900 × 10-6 / °C and the volumetric thermal expansion coefficient for the hydrometer glass 

is 26 ×10-6 / °C. What is the sp gr 60/60 of the oil? Assume that surface tension effects are 

already accounted for in the 0.7800 reading. 

Solution: Use Equation B8 to calculate the temperature correction for the hydrometer: 

( ) ' ( ) 0.7800 (25 C 15.56 C)(900 10 / C 26 10 / C) 0.0068 6 6

T = − − 0 L + g = − − × + × = − − −

′

o o o o c S T T β β

sp gr 60/60 F = S + cT′ = 0.7800 + ( ) − 0.0068 = 0.7732 o 

The correction of Equation B8 can be applied to other hydrometer scales that are functions of 

specific gravity T0 / T0 scales by converting them to specific gravity, making the necessary 

temperature corrections, and then returning them to the original scale. 

Example B3:

Given: An API gravity hydrometer reads 45.40 when floated in oil at 22 °C. The volumetric 

thermal expansion coefficient of the oil is known to be 800 × 10-6 / °C and the volumetric 

thermal expansion coefficient for the hydrometer glass is 26 ×10-6 / °C. What is the temperature 

corrected API gravity of the oil? Assume that surface tension effects are already accounted for in 

the 45.40 reading. 

Solution: First use the rearranged definition of API gravity to obtain the uncorrected sp gr 60/60 

°F reading for the oil: 
Hydrometer Calibration Service NIST SP 250-78

 Page 30 October 20, 2011 

0.7999

45.4 131.5

141.5

API Gravity, deg 131.5

141.5

sp gr 60/60 F = + = + = o . 

Calculate the temperature correction and the corrected specific gravity: 

( ) ' ( ) 0.7999 (22 C 15.56 C)(800 10 / C 26 10 / C) 0.0043 6 6

T = − − 0 L + g = − − × + × = − − −

′

o o o o c S T T β β

sp gr 60/60 F = S + cT′ = 0.7999 + ( ) − 0.0043 = 0.7956 o . 

Convert the corrected specific gravity back to API gravity: 

131.5 46.35

0.7956

141.5 131.5

sp gr 60/60 F

141.5 API Gravity, deg = − = − = o . 
Hydrometer Calibration Service NIST SP 250-78

 Page 31 October 20, 2011 

Appendix C. Conversion of Liquid Density to Other Hydrometer Scales [7, 4] 

Hydrometers calibrated in degrees Baumé at 20 °C are used to determine the percentage of cane 

sugar by weight in solution. The scale is defined by 

degrees Baumé at 20 °C = 145 – 145/(specific gravity 20 °C/20 °C). 

Expressed in terms of density, ρ (g/cm3

), the scale is defined by 

degrees Baumé at 20 °C = 145 – 145/[ρ(20 °C)/0.998206 g/cm3

]. 

The scale for indicating degrees Baumé of liquids lighter than water is defined by 

degrees Baumé = 140/( specific gravity 60 °F/60 °F) – 130. 

Expressed in terms of density, ρ, the scale is defined by 

degrees Baumé = 140/[ρ(60 °F)/0.999016 g/cm3

] – 130. 

The scale indicating degrees Baumé for liquids heavier than water is defined by 

degrees Baumé = 145 – 145/( specific gravity 60 °F/60 °F). 

Expressed in terms of density, ρ, the scale is defined by 

degrees Baumé = 145 – 145/[ρ(60 °F)/0.999016 g/cm3

]. 

The hydrometers in general use in the petroleum industry in the United States are based on the 

American Petroleum Institute (API) scale. The relationship between specific gravity and the API 

scale is 

degrees API = 141.5/( specific gravity 60 °F/60 °F ) – 131.5. 

Expressed in terms of density, ρ, the scale is defined by 

degrees API = 141.5/[ρ(60 °F)/0.999016 g/cm3

] – 131.5. 

 

[4] Standard Density and Volumetric Tables, NBS Circular 19, U.S. Government Printing Office, 

Washington D.C., 1924. 
Hydrometer Calibration Service NIST SP 250-78

 Page 32 October 20, 2011 

Appendix D. Nomenclature 

C hydrometer correction, s γ = c + c

CR hydrometer correction for a reference liquid R 

CL hydrometer correction for a particular liquid under test L 

s c scale correction 

γ c surface tension correction 

γR c surface tension correction for a reference liquid 

γL c surface tension correction for a particular liquid under test 

γ Δc change in surface tension correction between two liquids, γL γR = c − c

cT′ hydrometer correction for temperature effects 

B coefficient for surface tension correction from Cuckow’s method 

D hydrometer stem diameter 

g gravitational constant, = 980.665 cm/s2

m mass of a hydrometer 

R density hydrometer reading 

S specific gravity hydrometer reading 

sp gr 60/60 F

o specific gravity hydrometer scale, (60 F) (60 F) L H2Oo o = ρ ρ

T temperature 

T0 reference temperature 

T ′ temperature at which a hydrometer measurement is performed 

V submerged volume of a hydrometer 

ν volume of the hydrometer stem above the liquid surface 

VR submerged volume for a reference hydrometer 

β g volumetric thermal expansion coefficient for glass 

β L volumetric thermal expansion coefficient for liquid under test L 

L γ surface tension for liquid under test L 

R γ surface tension for reference liquid R 

ρ density 

ρ a1 density of air surrounding stem 

ρ L density of a liquid under test L 

ρ R density of a reference liquid R 

ρ H2O ( ) T density of water at temperature T, ( ) 3

H2O 60 F = 0.999016 g/cm o ρ

ρ mid density represented by the middle of the hydrometer range 

R+CR ρ density according to a reference hydrometer, without surface tension corrections 

θ L contact angle for liquid under test L 
NIST Test Number: 836-999999-07-05 Service ID No. 16010C 

Calibration Date: May 1, 2007 by Sherry Sheckels and John Wright Page 1 of 2

Appendix E. Sample Calibration Report 

REPORT OF CALIBRATION

FOR A 

HYDROMETER 

May 18, 2007

Maker: Hydrometer Co. Serial No.: 123456 

Indication: Specific Gravity 60/60 

Hydrometer Scale: 0.850 to 0.875 

submitted by 

Hydrometer User Co. 

124 Water Way 

Gaithersburg, MD 20899 

(Purchase Order Number 3456, dated April 17, 2007) 

The hydrometer identified above was calibrated by hydrostatic weighing (Cuckow’s method)1

 at three 

stem scale marks at approximately 10 %, 50 %, and 90 % of the hydrometer range. The calibration 

method requires weighing the hydrometer in air and while the hydrometer is suspended at the stem scale 

mark to be calibrated in a liquid of known density and surface tension. These data were used to calculate 

the density of the liquid (with zero surface tension) that would freely float the hydrometer at the 

calibrated stem scale marks. The Cuckow’s method density measurements were converted to the units 

displayed on the hydrometer under test (specific gravity 60/60) by dividing by the density of water at 60 

°F (0.999016 g/cm3

). 

The NIST calibration results are given as a correction to the hydrometer reading in the following form: 

L L L C = A + Bγ cosθ (1) 

where CL is the correction (for the customer’s liquid, L) to be added to the hydrometer reading, A and B

are calibration constants, L γ is the surface tension of the customer’s liquid (in dynes/cm), and θ L is the 

contact angle of the customer’s liquid on the hydrometer stem (normally assumed to be zero). The 

quantity A is the correction to the hydrometer for a liquid with zero surface tension and A has the same 

units as the hydrometer scale. The second term in Equation (1) corrects for the effects of hydrometer and 

liquid surface interactions and B has the units of the hydrometer scale units divided by dynes/cm. 

Typically, ignoring the second term will affect density measurements by about 0.2 % for aqueous 

solutions and about 0.1 % for hydrocarbon liquids. Table 1 lists the calibrated hydrometer marks, the 

values of A and B at each calibrated mark, the uncertainty of the density measurements made at NIST, and 

example hydrometer corrections from Equation (1) assuming a customer surface tension of 25 dynes/cm 

and a contact angle of zero. 

Table 1. Calibration results and example correction values for L γ = 25 dynes/cm. 

 

1 Aguilera, J., Wright, J., and Bean, V., Hydrometer Calibration by Hydrostatic Weighing with Automated 

Liquid Surface Positioning, Meas. Sci. Technol. 19 (2008) 015104.
REPORT OF CALIBRATION Hydrometer 

Hydrometer User Co. Purchase Order No: 3456 

NIST Test Number: 836-999999-07-05 Service ID No. 16010C 

Calibration Date: May 1, 2007 by Sherry Sheckels and John Wright Page 2 of 2 

Calibrated Mark A B Expanded Uncertainty2

(parts in 106) 

Example CL

0.8530 -0.00065 1.765E-05 100 -0.00020 

0.8630 -0.00049 1.784E-05 100 -0.00005 

0.8730 -0.00068 1.807E-05 100 -0.00023 

-0.00040

-0.00035

-0.00030

-0.00025

-0.00020

-0.00015

-0.00010

-0.00005

0.00000

0.00005

0.00010

0.850 0.855 0.860 0.865 0.870 0.875 0.880

Hydrometer Reading

Example Correction

1-May-07

19-May-75

6-Mar-70

7-Mar-66

Figure 1. Example hydrometer corrections along with the results of prior NIST calibrations (if available) 

The uncertainty of the results from the Cuckow’s method apparatus includes type B components related 

to balance measurements, the density and surface tension of the liquid used in the apparatus, the ability to 

align the hydrometer mark with the liquid surface, and the thermal expansion coefficient for the 

hydrometer glass. Type A uncertainty is assessed from five repeated measurements made at the each 

calibrated mark. 

When the hydrometer is applied by the customer to measure the density of a liquid, uncertainties beyond 

the NIST calibration must be considered. These uncertainty components include: the surface tension of 

the liquid under test, the cleanliness of the hydrometer stem and its effect on the contact angle of the 

liquid with the hydrometer stem, thermometer calibration, and temperature non-uniformity of the liquid 

sample. When these uncertainty components are considered, the uncertainty of the liquid density 

measurement made with the hydrometer can be 400 parts in 106

 or larger.1

For the Director, 

Dr. John D. Wright 

Project Leader, Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratory 

National Institute of Standards and Technology 

Sherry Sheckels 

Calibration Technician, Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratory 

National Institute of Standards and Technology 

 

2 Approximately 95 % confidence level (k = 2), see Taylor, B. N. and Kuyatt, C. E., Guidelines for 

Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, 

NIST (January, 1993).