Question

An insulated cylinder is initially divided into halves by a frictionless, thermally conducting piston. On one side of the piston is \(1 \mathrm{~m}^{3}\) of a gas at \(300 \mathrm{~K}, 2\) bar. On the other side is \(1 \mathrm{~m}^{3}\) of the same gas at \(300 \mathrm{~K}, 1\) bar. The piston is released and equilibrium is attained, with the piston experiencing no change of state. Employing the ideal gas model for the gas, determine (a) the final temperature, in \(\mathrm{K}\). (b) the final pressure, in bar. (c) the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{kg}\).

Short answer

The final temperature is 300 K, the final pressure is 1.5 bar, and the entropy produced is 0.002005 kJ/K.

Step-by-step solution

Understanding Initial Conditions

Identify the initial conditions on both sides of the piston. On the left side, we have 1 m³ of the gas at 300 K and 2 bar. On the right side, we have 1 m³ of the same gas at 300 K and 1 bar.

Applying Ideal Gas Law

Using the ideal gas law, calculate the initial number of moles on each side. The ideal gas law is \[pV = nRT\] where \( p \) is pressure, \( V \) is volume, \( n \) is number of moles, \( R \) is the ideal gas constant (\( R = 0.08314 \) J/(mol·K)), and \( T \) is temperature.

Calculating Initial Moles

Calculate the number of moles on the left side:\[ n_1 = \frac{p_1 V_1}{RT} = \frac{2 \times 1}{0.08314 \times 300} = 0.08 \text{ moles} \]On the right side:\[ n_2 = \frac{p_2 V_2}{RT} = \frac{1 \times 1}{0.08314 \times 300} = 0.04 \text{ moles} \]

Finding Final Pressure

At equilibrium, the final pressure \( p_f \) will be uniform throughout the cylinder. Use the total number of moles (\(n_1 + n_2 = 0.08 + 0.04 = 0.12 \, \text{moles} \)) and the total volume (2 m³) to find \(p_f\), since temperature (T) is constant.\[ p_f = \frac{n_f RT}{V_f} = \frac{0.12 \times 0.08314 \times 300}{2} = 1.5 \, \text{bar} \]

Determining Final Temperature

Since the piston is thermally conducting and there is no heat transfer to the surroundings, the final temperature must be the same as the initial temperature, which is \( T_f = 300 \text{K} \).

Calculating Entropy Produced

The entropy change for each side using the Gibbs equation \( \triangle S = nR \ln\left(\frac{V_f}{V_i}\right) \). First for the side initially at 2 bar:\[ \triangle S_1 = 0.08 \times 0.08314 \ln(2) = 0.004009 \, \text{kJ/K} \] Then for the side initially at 1 bar:\[ \triangle S_2 = 0.04 \times 0.08314 \ln(0.5) = -0.002004 \text{kJ/K} \] The net entropy change is:\[ \triangle S = \triangle S_1 + \triangle S_2 = 0.004009 - 0.002004 = 0.002005 \, \text{kJ/K} \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is fundamental to understanding thermodynamics. It relates the pressure (p), volume (V), and temperature (T) of an ideal gas to the amount of gas present measured in moles (n). This relationship is given by the formula: \[ pV = nRT \] where R is the ideal gas constant, which has a value of 0.08314 J/(mol·K). The law indicates that if you know any three of the properties (p, V, T, n), you can calculate the fourth. This principle is crucial when analyzing the initial states of the gas on both sides of the piston, as we do in steps to find the number of moles present initially.

Entropy Change

Entropy is a measure of disorder or randomness in a system. In thermodynamics, changes in entropy (ΔS) can tell us about the energy dispersal in a process. For an ideal gas, the entropy change can be calculated using the formula: \[ \triangle S = nR \ln\left(\frac{V_f}{V_i}\right) \] Here, n is the number of moles, V_f is the final volume, V_i is the initial volume, and R is the gas constant. In the problem, we calculate the entropy change for both sides of the piston separately and then add them to find the total entropy produced in the system. It's important to note that even though one side experiences an increase in entropy, the other side's entropy decreases, contributing to the total system's final entropy state.

Thermodynamic Equilibrium

Thermodynamic equilibrium is a state where all macroscopic flows of energy and matter have ceased and the properties of the system (e.g., pressure, temperature) are uniformly distributed throughout. In this problem, when the piston is released, the gas moves until there is no further net force, reaching equilibrium. At equilibrium, the pressure on both sides of the piston becomes equal. This final pressure can be found using the ideal gas law across the total volume and total number of moles. Understanding equilibrium helps explain why the final pressure in the cylinder is uniform despite the initial differences.

Insulated Systems

An insulated system is one where no heat can enter or leave; it is thermally isolated from its surroundings. This concept is key in the given problem because it implies the internal energy changes due to work done by the gases without any energy exchange with the environment. The problem states that the cylinder is insulated and the piston is thermally conducting, meaning heat can only be exchanged between the compartments of the gas. This isolation ensures the final temperature remains the same as the initial temperature, simplifying the calculation of final states and entropy changes.