(i) Show that if the molar heat capacity in the ideal gas...

(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 $$ \begin{aligned} F(T, V)= & U_0-T S_0+\frac{3}{2} R\left(T-T_0\right) \\ & -\left(I(V)-I\left(V_0\right)\right)(1+x(\omega, T))^2 \\ & -R T \ln \left[\frac{V-b}{V_0-b}\left(\frac{T}{T_0}\right)^{3 / 2}\right] \end{aligned} $$

(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
$$
\begin{aligned}
F(T, V)= & U_0-T S_0+\frac{3}{2} R\left(T-T_0\right) \\
& -\left(I(V)-I\left(V_0\right)\right)(1+x(\omega, T))^2 \\
& -R T \ln \left[\frac{V-b}{V_0-b}\left(\frac{T}{T_0}\right)^{3 / 2}\right]
\end{aligned}
$$

Key Concepts

Molar Heat Capacity

Molar heat capacity is a thermodynamic property that measures the amount of heat required to raise the temperature of one mole of a substance by one degree. In the ideal gas limit, this property is often derived from the kinetic theory of gases, resulting in the well?known value of (3/2)R for monoatomic gases. Understanding molar heat capacity is essential for evaluating how energy is stored in the degrees of freedom of a system and how it changes with temperature, especially when comparing idealized behavior to that of real gases.

Peng-Robinson Equation of State

The Peng-Robinson equation of state is a cubic model widely used to describe the behavior of real gases. It incorporates terms that account for the attractive and repulsive intermolecular forces in fluids, offering corrections beyond the ideal gas law. This equation includes parameters such as a and b, which are related to the specific molecular interactions, as well as parameters that depend on the acentric factor. This formulation is crucial for accurately predicting properties like pressure, volume, and temperature and for extending thermodynamic descriptions from ideal gases to more realistic scenarios.

Helmholtz Free Energy

The Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed system at constant temperature and volume. It serves as a bridge between microscopic energy contributions and macroscopic observables, playing a central role in determining equilibrium states. In the context of real gases, expressing the Helmholtz free energy by integrating relevant thermodynamic expressions allows one to incorporate non-ideal behavior and corrections provided by models like the Peng-Robinson equation of state.

Thermodynamic Integration

Thermodynamic integration is a mathematical technique used to derive changes in thermodynamic properties by integrating differential relationships from a reference state to the state of interest. In problems like this, integration is used to account for non-ideal contributions, such as those arising from intermolecular forces, by integrating functions like I(V) over the volume. This method is an essential tool for relating changes in energy, pressure, and volume when direct measurement or algebraic solutions are not straightforward.

Transcript

00:01 Hello everyone here it is given a certain ideal gas has molar heat capacity at constant volume so cv is molar specific heat at constant volume assuming this gas initially occupies a volume v0 so initial volume is v0 at pressure p .0 and temperature t not kelvin the there's expand isobarically so there is a process isobaric expansion so that volume becomes in this process is to be not and then expand further so this is the first process and second process there is an aerobatic expansion so that volume becomes 4b0 we have to draw pv graph for the sequence of process.

02:03 Pv graph, second part, we have to compute the work done by the gas for the sequence of process, work done by the gas.

02:20 See, we have to find the final temperature, that is t3, find the absolute value q, amount of heat transfer in the process.

02:44 Now let us start solving it.

02:52 Now pv graph first pv graph first process is isobaric that is temperature is constant first process is 1 to 2 in which pressure is constant 2 to 3 is is isobaric sorry adiabetic expansion now second part b this is the answer of part a answer of p part we have to calculate work done so work done in the first process initial pressure pressure into change in volume that is p not final volume 2 v0 minus initial volume so it will be p .0 v0 in second process work done cb upon r p2 b2 minus p3 v3 substitute the value so it becomes c b r p2 is p0 2 v0 minus p3 into 4v0 but b3 we don't know so b3 can be calculated here here b3 having the value sorry p3 p2 into v2 b2 upon b1 to the power gamma so it is p1 2 b0 upon 4 b0 to the power gamma substitute this value here.

05:58 So w2 you may write this is your first cv by r to p not b not minus p not 2 to the power minus gamma.

06:30 So total work done in the process w1 plus w2 p.

06:39 Not b0 plus c v .r this is p3 i...

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