Problem 24 - A Modern Course in Statistical Physics

Question

A van der Waals gas can be cooled by free expansion. Since no work is done and no heat is added during free expansion, the internal energy remains constant. An infinitesimal change in volume \(\mathrm{d} V\) causes an infinitesimal temperature change in \(\mathrm{d} T\), where $$ \mathrm{d} T=\left(\frac{\partial T}{\partial V}\right)_{U, \mathrm{n}} \mathrm{d} V $$ (a) Compute the Joule coefficient \({ }^{(\partial T / \partial V)}_{U, \mathrm{H}}\) for a van der Waals gas (note that the heat capacity \(C_{V, n}\) is independent of volume and use \(C_{V, n}=3 / 2 \mathrm{n} R\) ) (b) Compute the change in temperature of one mole of oxygen \(\left(\mathrm{O}_{2}\right)\) and one mole of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) if they each expand from an initial volume \(V_{\mathrm{i}}=10^{-3} \mathrm{~m}^{3}\) at temperature \(T_{\mathrm{i}}=300 \mathrm{~K}\) to a final volume \(V_{\mathrm{f}}=\infty\). (For \(\mathrm{O}_{2}\) the van der Waals constant is \(a=0.1382 \mathrm{~Pa} \mathrm{~m}^{6} / \mathrm{mol}^{2}\) and for \(\mathrm{CO}_{2}\) it is \(a=0.3658 \mathrm{~Pa} \mathrm{~m}^{6} / \mathrm{mol}^{2}\).)

Step-by-Step Solution

Answer

The temperature change for 1 mole of \( O2 \) is 11.0716 K and for 1 mole of \(CO2) it is 29.69 K.

Step 1: Identify the Joule coefficient formula

The Joule coefficient is defined as \ \ \ \( \left( \frac{\partial T}{\partial V} \right)_{U, n} \). \ For the van der Waals equation, the internal energy is related to the term \( a \) as such a gas cannot be treated as ideal.

Step 2: Use energy relation for van der Waals gas

For a van der Waals gas, the internal energy U is a function of T and V. We use: \ \ \ \( U = \frac{3}{2} nR T - \frac{a n^2}{V} \). \ \ Taking \( dU = 0 \) because energy remains constant, leads to: \ \ \( \frac{3}{2} n R dT - \frac{an^2}{V^2} dV = 0 \).

Step 3: Solve for the partial derivative \( \left( \frac{\partial T}{\partial V} \right)_{U, n} \)

From \( \frac{3}{2} n R dT = \frac{an^2}{V^2} dV \), dividing both sides by \( dV \) and isolating \( \frac{dT}{dV} \), we get: \ \ \( \left( \frac{\partial T}{\partial V} \right)_{U, n} = \frac{2an}{3RV^2} \).

Step 4: Calculate the temperature change for O2

Given: \( n = 1 \) mole, \( C_{V, n} = \frac{3}{2} nR \), starting from Eq \( \Delta T = \int_{Vi}^{Vf} \left( \frac{\partial T}{\partial V} \right)_{U, n} dV \):\ floowing this: \( \Delta T = \int_{0.001}^{∞} \frac{2 \times 0.1382}{3 \times 8.31446 \times V^2} dV \). Solving this, we get: \( \Delta T = [ \frac{-2 \times 0.1382}{3 \times 8.31446 \times V} ]_{0.001}^{∞} = [ 0 - (-11.0716) ] = 11.07 K \)

Step 5: Calculate the temperature change for CO2

Similarly, for \( CO2 \): \( \Delta T = \int_{0.001}^{∞} \frac{2 \times 0.3658}{3 \times 8.31446 \times V^2} dV \). Solving this, we get: \( \Delta T = [ \frac{-2 \times 0.3658}{3 \times 8.31446 \times V} ]_{0.001}^{∞} = [ 0 - (-29.69) ] = 29.69 K \)

Key Concepts

Joule Coefficient

The Joule coefficient is a thermodynamic property that measures the change in temperature of a gas during free expansion when no heat is added or removed. It is represented by \(\frac{\partial T}{\partial V}\)_{U, n}. Free expansion implies that no work is done and the internal energy remains constant.\ Using the van der Waals equation, which accounts for the non-ideal behavior of gases due to molecular attractions and finite molecular sizes, the internal energy U can be expressed as: \[ U = \frac{3}{2} nR T - \frac{a n^2}{V} \] Here, a is the van der Waals constant representing intermolecular forces.\ To compute the Joule coefficient for a van der Waals gas, we set the infinitesimal change in internal energy dU to zero and solve for \( \frac{dT}{dV} \):\ \[ \frac{3}{2} n R \frac{dT}{dV} = \frac{an^2}{V^2} \]\ Upon isolating \( \frac{dT}{dV} \), we get: \[ \left( \frac{\partial T}{\partial V} \right)_{U, n} = \frac{2an}{3RV^2} \] Understanding this relationship helps in predicting temperature changes under conditions of free expansion for real gases.

Internal Energy

Internal energy is a critical concept in thermodynamics that refers to the total energy contained within a system due to the kinetic and potential energies of its molecules. In the context of van der Waals gases, internal energy is influenced not only by temperature but also by volume and intermolecular forces.\ For an ideal gas, internal energy is solely a function of temperature. However, real gases deviate from this ideal behavior, which is captured by the van der Waals equation. The internal energy U for a van der Waals gas is expressed as: \[ U = \frac{3}{2} nR T - \frac{a n^2}{V} \]\ The term \( \frac{3}{2} nR T \) represents the kinetic energy of the gas molecules, while \( -\frac{a n^2}{V} \) accounts for the potential energy due to intermolecular attractions. During free expansion, no work is done on or by the gas, and the internal energy remains constant, which is a crucial consideration in calculating temperature changes.

Van der Waals Equation

The van der Waals equation is a modified version of the ideal gas law that accounts for the non-ideal behavior of real gases. It introduces two parameters: a and b, which correct for intermolecular attractions and the finite size of gas molecules, respectively. The equation is expressed as: \[ \left( P + \frac{a n^2}{V^2} \right) (V - nb) = nRT \]\ Here: \[n \] is the number of moles,\ \[P \] is the pressure,\ \[V \] is the volume,\ \[R \] is the gas constant,\ \[a \] is a measure of the attraction between particles,\ \[b \] is the volume occupied by the gas particles themselves.\ This equation adjusts the ideal gas law \( PV = nRT \) to more accurately reflect the behavior of real gases. The correction term \( \frac{a n^2}{V^2} \) reduces the pressure to account for attractive forces, while \( nb \) accounts for the volume occupied by the gas molecules, thus reducing the effective volume available for gas particles to move in. Understanding the van der Waals equation is essential for predicting the behavior of gases under various conditions, such as high pressures and low temperatures, where deviations from ideal behavior are significant.