Subfield: Thermodynamics

Van der Waals equation of state

In a well-known model of an ideal gas, whose equation of state obeys the Clapeyron-Mendeleev law, the following important physical effects are neglected. First, molecules of a real gas have a finite size and, secondly, they interact with one another. In all parts of this problem one mole of water is considered.

Part A. Non-ideal gas equation of state

Taking into account the finite size of the molecules, the gaseous equation of state takes the form

$$
P(V-b)=R T
\tag{1}
$$

where $P, V, T$ stands for the gas pressure, its volume per mole and temperature, respectively, $R$ denotes the universal gas constant, and $b$ is a specific constant extracting some volume.
Context question:
A1 Estimate $b$ and express it in terms of the diameter of the molecules $d$.
Context answer:
\boxed{$b=N_{A} d^{3}$}


Extra Supplementary Reading Materials:

With account of intermolecular attraction forces, van der Waals proposed the following equation of state that neatly describes both the gaseous and liquid states of matter

$$
\left(P+\frac{a}{V^{2}}\right)(V-b)=R T
\tag{2}
$$

where $a$ is another specific constant.

At temperatures $T$ below a certain critical value $T_{c}$ the isotherm of equation (2) is well represented by a non-monotonic curve 1 shown in Figure 1 which is then called van der Waals isotherm. In the same figure curve 2 shows the isotherm of an ideal gas at the same temperature. A real isotherm differs from the van der Waals isotherm by a straight segment $\boldsymbol{A B}$ drawn at some constant pressure $P_{L G}$. This straight segment is located between the volumes $V_{L}$ and $V_{G}$, and corresponds to the equilibrium of the liquid phase (indicated by $L$ ) and the gaseous phase (referred to by $G$ ). From the second law of thermodynamics J. Maxwell showed that the pressure $P_{L G}$ must be chosen such that the areas $I$ and $I I$ shown in Figure 1 must be equal.

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Figure 1. Van der Waals isotherm of gas/liquid (curve 1) and the isotherm of an ideal gas (curve 2).

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Figure 2. Several isotherms for van der Waals equation of state.

With increasing temperature the straight segment $A B$ on the isotherm shrinks to a single point when the temperature and the pressure reaches some values $T_{c}$ and $P_{L G}=P_{c}$, respectively. The parameters $P_{c}$ and $T_{c}$ are called critical and can be measured experimentally with high degree of accuracy.
Context question:
A2 Express the van der Waals constants $a$ and $b$ in terms of $T_{c}$ and $P_{c}$.
Context answer:
\boxed{$a=\frac{27 R^{2} T_{c}^{2}}{64 P_{c}}$ , $b=\frac{R T_{c}}{8 P_{c}}$}


Context question:
A3 For water $T_{c}=647 \mathrm{~K}$ and $P_{c}=2.2 \cdot 10^{7} \mathrm{~Pa}$. Calculate $a_{w}$ and $b_{w}$ for water.
Context answer:
\boxed{$a_{w}=0.56$ , $b_{w}=3.1 \cdot 10^{-5} $}


Context question:
A4 Estimate the diameter of water molecules $d_{w}
Context answer:
\boxed{$4 \cdot 10^{-10}$}


Extra Supplementary Reading Materials:

Part B. Properties of gas and liquid

This part of the problem deals with the properties of water in the gaseous and liquid states at temperature $T=100{ }^{\circ} \mathrm{C}$. The saturated vapor pressure at this temperature is known to be $p_{L G}=p_{0}=1.0$. $10^{5} \mathrm{~Pa}$, and the molar mass of water is $\mu=1.8 \cdot 10^{-2} \frac{\mathrm{kg}}{\mathrm{mole}}$.

Gaseous state

It is reasonable to assume that the inequality $V_{G} \gg b$ is valid for the description of water properties in a gaseous state.
Context question:
B1 Derive the formula for the volume $V_{G}$ and express it in terms of $R, T, p_{0}$, and $a$.
Context answer:
\boxed{$V_{G}=\frac{R T}{2 p_{0}}(1+\sqrt{1-\frac{4 a p_{0}}{R^{2} T^{2}}})$}


Extra Supplementary Reading Materials:

Almost the same volume $V_{G 0}$ can be approximately evaluated using the ideal gas law.
Context question:
B2 Evaluate in percentage the relative decrease in the gas volume due to intermolecular forces, $\frac{\Delta V_{G}}{V_{G 0}}=\frac{V_{G 0}-V_{G}}{V_{G 0}}$.
Context answer:
\boxed{0.58 %}


Extra Supplementary Reading Materials:

If the system volume is reduced below $V_{G}$, the gas starts to condense. However, thoroughly purified gas can remain in a mechanically metastable state (called supercooled vapor) until its volume reaches a certain value $V_{G \mathrm{~min}}$.

The condition of mechanical stability of supercooled gas at constant temperature is written as: $\frac{d P}{d V}<$ 0.
Context question:
B3 Find and evaluate how many times the volume of water vapor can be reduced and still remains in a metastable state. In other words, what is $V_{G} / V_{G \text { min }}$ ?
Context answer:
\boxed{86}


Extra Supplementary Reading Materials:

Liquid state

For the van der Waals' description of water in a liquid state it is reasonable to assume that the following inequality holds $P \ll a / V^{2}$.
Context question:
B4 Express the volume of liquid water $V_{L}$ in terms of $a, b, R$, and $T$.
Context answer:
\boxed{$V_{L}=\frac{a}{2 R T}(1-\sqrt{1-\frac{4 b R T}{a}})$}


Extra Supplementary Reading Materials:

Assuming that $b R T \ll a$, find the following characteristics of water. Do not be surprised if some of the data evaluated do not coincide with the well-known tabulated values!
Context question:
B5 Express the liquid water density $\rho_{L}$ in some of the terms of $\mu, a, b, R$ and evaluate it.
Context answer:
\boxed{$\rho_{L}=\frac{\mu}{b\left(1+\frac{b R T}{a}\right)}$ , $5.8 \cdot 10^{2} $}


Context question:
B6 Express the volume thermal expansion coefficient $\alpha=\frac{1}{V_{L}} \frac{\Delta V_{L}}{\Delta T}$ in terms of $a, b, R$, and evaluate it.
Context answer:
\boxed{$\alpha=\frac{b R}{a+b R T}$ , $4.6 \cdot 10^{-4}$}


Question: B7 Express the specific heat of water vaporization $L$ in terms of $\mu, a, b, R$ and evaluate it.

Solution:
The heat, required to convert the liquid to gas, is used to overcome the intermolecular forces that create negative pressure $a / V^{2}$, therefore,

$$
E=L \mu \approx \int_{V_{L}}^{V_{G}} \frac{a}{V^{2}} d V=a\left(\frac{1}{V_{L}}-\frac{1}{V_{G}}\right)
\tag{B7.1}
$$

and using $V_{G} \gg V_{L}$, (B7.1) yields

$$
L=\frac{a}{\mu V_{L}}=\frac{a}{\mu b\left(1+\frac{b R T}{a}\right)} \approx \frac{a}{\mu b}=1.0 \cdot 10^{6} \frac{\mathrm{J}}{\mathrm{kg}}
\tag{B7.2}
$$

Final answer: $L=\frac{a}{\mu b(1+\frac{b R T}{a})}$ , $1.0 \cdot 10^{6}$ $\frac{\mathrm{J}}{\mathrm{kg}}$