# Theory of Heat (Maxwell, 4th edition)/Chapter 6

CHAPTER VI.
ON LINES OF EQUAL TEMPERATURE, OR ISOTHERMAL LINES ON THE INDICATOR DIAGRAM.
If the pressure is made to vary while the temperature remains constant, the volume will diminish as the pressure increases, and the point P will trace out a line in the diagram which is called a line of equal temperature, or an isothermal line. By means of this line we can show the whole behaviour ​of the substance under various pressures at that particular temperature.
By making experiments on the substance at other temperatures, and drawing the isothermal lines belonging to these temperatures, we can express all the relations between the pressure, volume, and temperature of the substance.
In the diagram, each isothermal line should be marked with the temperature to which it corresponds in degrees, and the lines should be drawn for every degree, or for every ten or every hundred degrees, according to the purpose for which the diagram is intended.
When the volume and the pressure are known, the temperature is a determinate quantity, and it is easy to see how from any two of these three quantities we can determine the third. Thus if the curved lines in the diagram are the lines of equal temperature, the temperature cor. responding to each being indicated by the numeral at the end of the line, we can solve three problems by means of this diagram.
1. Given the pressure and the volume, to find the temperature.
Lay off O L on the line of volumes to represent the given volume, and O F on the line of pressures to represent the given pressure, then draw F P horizontal and L P vertical, to determine the point P. If the point P falls on one of the lines of equal temperature, the numeral attached to that line indicates the temperature. If the point P falls between two of the lines, we must estimate its distance from the two nearest lines, and then as the sum of these distances is to the distance from the lower line of temperature, so is the difference of temperature of the two lines to the excess of the true temperature above that of the lower line.
2. Given the volume and temperature to find the pressure.
Lay off O L to represent the volume and draw L P vertical, and let P be the point where this line cuts the line of the ​given temperature. Then L P represents the required pressure.
3. Given the pressure and temperature, to find the volume.
Fig. 13.
Lay off O F to represent the pressure and draw F P horizontal till it meets the line of the given temperature in P, then F P represents the required volume.
ON THE FORM OF THE ISOTHERMAL CURVES IN DIFFERENT CASES.
The Gaseous State.
If the substance is in the gaseous state, as common air is at every pressure and temperature to which we have been able to subject it, then it is easy to draw the isothermal curves by taking account of the laws of Boyle and Charles.
By Boyle's law the product of the volume and the pres​sure is always the same for the same temperature. Hence, in the curve, the area of the rectangle O L P F will be the same provided P be a point in the same isothermal curve.
The curve which has this property is known in geometry by the name of the rectangular hyperbola, the lines O$v$ and O$p$ being the asymptotes of the hyperbolas in fig. 13. The asymptotes are lines such that a point travelling along the curve in either direction continually approaches one or other of the asymptotes, but never reaches it. The physical interpretation of this is that if a gas fulfils Boyle's law, and if the temperature remain the same—
1. Suppose we travel along the curve in the direction leading toward O$p$, that is to say, suppose the pressure is gradually increased, then the volume will continually diminish, but always slower and slower; for, however much we increase the pressure, we can never reduce the volume to nothing, so that the isothermal line will never reach the line O$p$, though it continually approaches it. At the same time, if Boyle's law is fulfilled we can always, by doubling the pressure, reduce the volume to one half, so that by a sufficient increase of pressure the volume may be reduced till it is smaller than any prescribed quantity.
2. Suppose we travel in the other direction along the curve, that is to say, suppose we increase the volume of the vessel which contains the gas, then the point $p$ approaches nearer and nearer to the line O$v$, but never actually reaches it. This shows that the gas will always expand so as to fill the vessel, and press upon it with a force represented by the distance from O$v$, and this pressure, though it diminishes as the vessel is enlarged, will never be reduced to nothing, however large the vessel may become.
Elasticity of a Perfect Gas.—Another property of the hyperbola is that if P E be drawn a tangent to the curve at P till it meets the asymptote, F E = O F. Now F E represents the elasticity of the substance, and O F the pressure. Hence the elasticity of a perfect gas is numerically ​equal to the pressure, when the temperature is supposed to remain constant during the compression.
The Liquid State.
In most liquids, the compression produced by the pressures which we are able to apply is exceedingly small. In the case of water, for example, under ordinary circumstances as to temperature, the application of a pressure equal to one atmosphere produces a compression of about 46 millionth parts of the volume, or 0.000046. Hence in drawing an indicator diagram for a liquid we must represent changes of volume on a much larger scale than in the case of gases, if the diagram is to have any visible features at all. The most convenient way is to suppose the line O L to represent, not the whole volume, but the excess of the volume above a thousand or a million of the units we employ.
It is manifest that the relation between the pressure and the volume of any substance must be such that no pressure, however great, can reduce the volume to nothing. Hence the isothermal lines cannot be straight lines, for a straight line, however slightly inclined to the line of no volumes O F, and however distant from it, must cut that line somewhere. The limited range of pressures which we are able to produce does not in some cases cause sufficient change of volume to indicate the curvature of the isothermal lines. We may suppose that for the small portion we are able to observe they are nearly straight lines.
The expansion due to an increase of temperature is also much smaller in the case of liquids than in the case of gases.
If, therefore, we were to draw the indicator diagram of a liquid on the same scale as that of a gas, the isothermal lines would consist of a number of lines very close together, nearly vertical, but very slightly inclined towards the line O F.
If, however, we retain the scale of pressures and greatly magnify the scale of volumes, the isothermal lines will be ​more inclined to the vertical and wider apart, but still very nearly straight lines. Liquids, however, which are near the critical point described at the end of this chapter are more compressible than even a gas.
The Solid State.
In solid bodies the compressibility and the expansion by heat are in general smaller than in liquids. Their indicator diagrams will therefore have the same general characteristics as those of liquids.
INDICATOR DIAGRAM OF A SUBSTANCE PART OF WHICH IS LIQUID AND PART VAPOUR.
Let us suppose that a pound of water is placed in a vessel and brought to a given temperature, say 212° F., and that by means of a piston the capacity of the vessel is made larger or smaller, the temperature remaining the same. If we suppose the vessel to be originally very large, say 100 cubic feet, and to be maintained at 212° F., then the whole of the water will be converted into steam, which will fill the vessel and will exert on it a pressure of about 575 pounds' weight on the square foot. If we now press down the piston, and so cause the capacity of the vessel to diminish, the pressure will increase nearly in the same proportion as the volume diminishes, so that the product of the numbers representing the pressure and volume will be nearly constant. When, however, the volume is considerably diminished, this product begins to diminish, that is to say, the pressure does not increase so fast as it ought to do by Boyle's law if the steam were a perfect gas. In the diagram, fig. 14, p. 114, the relations between the pressure and volume of steam at 212° are indicated by the curve $a\ b$. The pressure in atmospheres is marked on the right hand of the diagram, and the volume of one pound, in cubic feet, at the bottom.
Fig. 14.
Isothermals for Steam and Water. ​pressure is 2,116 lb., so that the product of the volume and pressure, instead of 57,500, is now reduced to 55,770. This departure from the law of Boyle, though not very large, is quite decided. The pressure and volume of the steam in this state are indicated by the point $b$ in the diagram.
If we now diminish the volume and still maintain the same temperature, the pressure will no longer increase, but part of the steam will be converted into water; and as the volume continues to diminish, more and more of the steam will be condensed into the liquid form, while the pressure remains exactly the same, namely, 2,116 pounds' weight on the square foot, or one atmosphere. This is indicated by the horizontal line $b\ c$ in the diagram.
This pressure will continue the same till all the steam is condensed into water at 212°, the volume of which will be 0.016 of a cubic foot, a quantity too small to be represented clearly in the diagram.
As soon as the volume, therefore, is reduced to this value there will be no more steam to condense, and any further reduction of volume is resisted by the elasticity of water, which, as we have seen, is very large compared with that of a gas.
We are now able to trace the isothermal line for water corresponding to the temperature 212°. When V is very great the curve is nearly of the form of an hyperbola for which V P = 57,500. As V diminishes, the curve falls slightly below the hyperbola, so that when V = 26.36, V P = 55,770. Here, however, the line suddenly and completely alters its character, and becomes the horizontal straight line bc, for which P = 2,116, and this straight line extends from V = 26.36 to V = 0.016, when another equally sudden change takes place, and the line, from being exactly horizontal, becomes nearly but not quite vertical, nearly in the direction $c\ p$, for the pressure must be increased beyond the limits of our experimental methods long before any very considerable change is made in the volume of the water. ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/132 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/133 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/134 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/135 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/136 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/137 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/138 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/139 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/140 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/141 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/142 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/143