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Thermodynamics: A complete undergraduate course

Andrew M. Steane

Chapter 15

Modelling real gases - all with Video Answers

Educators

Chapter Questions

32:10

Problem 1

Show that for a van der Waals gas,
$$
\left.\frac{\partial C_V}{\partial V}\right|_T=0, \quad C_p-C_V=\frac{R}{1-2 a(V-b)^2 /\left(R T V^3\right)} .
$$

Shalini Tyagi
Shalini Tyagi
Numerade Educator
01:01

Problem 2

(i) Show that for both a van der Waals gas and a Dieterici gas, the $B_2$ coefficient in the virial expansion is given by
$$
B_2=b-\frac{a}{R T}
$$
Hence show that the Boyle temperature is 3.375 times the critical temperature for the van der Waals gas, and 4 times for the Dieterici gas. (ii) Find the next virial coefficient for both these equations of state.

Narayan Hari
Narayan Hari
Numerade Educator
04:56

Problem 3

The van der Waals equation is a cubic equation in $V$, having either three real roots or just one real root at any given $p, T$. For $T<T_c$ and $p<p_c$ there are three real roots. As $T$ increases, these roots approach one another and merge at $T=T_c$. Therefore at the critical point the equation of state must take the form $\left(V-V_c\right)^3=0$. Use this to find the critical parameters in terms of $a$ and $b$, by expanding the bracket and comparing with the van der Waals equation.

Farhana Sharmin
Farhana Sharmin
Numerade Educator
02:22

Problem 4

Using the method introduced in question 15.3, or otherwise, show that for the Redlich-Kwong equation, the critical compression factor $\left(p_c V_c / R T_c\right)$ is $Z_c=1 / 3$, and the critical volume is $V_c=x b$, where $x$ is a solution of the cubic equation
$$
x^3-3 x^2-3 x-1=0
$$
Confirm (e.g. by plotting the function) that this function has one real root, at $x=1+2^{1 / 3}+2^{2 / 3} \simeq 3.84732$. Show that the critical temperature is $T_c=\left(3 a / x^2 R b\right)^{2 / 3}$ and obtain equation (15.20).

Gregory Higby
Gregory Higby
Numerade Educator
03:54

Problem 5

Using equation (13.31), show that the constant volume heat capacity of a Redlich-Kwong gas is given by
$$
C_V(T, V)=C_V(T)^{\text {(udeal })}-\frac{R}{4 \beta^2} t^{-3 / 2} \ln \frac{v}{v+\beta},
$$
where $v=V / V_c, t=T / T_c$, and $\beta=\left(1+2^{1 / 3}+2^{2 / 3}\right)^{-1}$. Hence find the fractional reduction in $C_V$, compared to the ideal gas, at $t=1, v=2$, when $C_V^{(\text {dideal) }}=3 R / 2$.

Mukesh Devi
Mukesh Devi
Numerade Educator
11:00

Problem 6

(i) Show that if the molar heat capacity in the ideal gas limit is $C_V=$ $(3 / 2) R$, then the molar heat capacity of a Peng-Robinson gas is
$$
C_V=\frac{3}{2} R+\frac{y(y+1)\left(I(V)-I\left(V_0\right)\right)}{2\left(T T_c\right)^{1 / 2}},
$$
where $y=0.37464+1.54226 \omega-0.26992 \omega^2$ and
$$
\begin{aligned}
I(V) & \equiv-\int \frac{a \mathrm{~d} V}{V^2+2 V b-b^2} \\
& =\frac{a}{2 \sqrt{2} b} \ln \left[\frac{V+(1+\sqrt{2}) b}{V+(1-\sqrt{2}) b}\right] .
\end{aligned}
$$
(ii) Show that the molar Helmholtz function is given by
$$
\begin{aligned}
F(T, V)= & U_0-T S_0+\frac{3}{2} R\left(T-T_0\right) \\
& -\left(I(V)-I\left(V_0\right)\right)(1+x(\omega, T))^2 \\
& -R T \ln \left[\frac{V-b}{V_0-b}\left(\frac{T}{T_0}\right)^{3 / 2}\right]
\end{aligned}
$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:46

Problem 7

In Yournal of Chemical Physics vol. 115 (2001), R. Sadus proposes the two-parameter equation of state
$$
p V=R T \frac{1+y+y^2-y^3}{(1-y)^3} e^{-a / R T V}
$$
where $y=b / 4 V$ and $a, b$ are parameters. Considering that it has only two empirically determined parameters, this equation of state shows a good ability to predict the density of both liquid and vapour phases and to locate the phase boundary. Show that $y=0.357057$ at the critical point and find the critical compression factor.

Manik Pulyani
Manik Pulyani
Numerade Educator