Consider a perfect gas contained in a cylinder and separated by a frictionless adiabatic piston into two sections \(\mathrm{A}\) and \(\mathrm{B}\). All changes in \(\mathrm{B}\) is isothermal; that is, a thermostat surrounds \(\mathrm{B}\) to keep its temperature constant. There is \(2.00 \mathrm{~mol}\) of the gas in each section. Initially, \(T_{\mathrm{A}}==T_{\mathrm{B}}=300 \mathrm{~K}, V_{\mathrm{A}}=\) \(V_{\mathrm{B}}\) \(=2.00 \mathrm{dm}^{3}\). Energy is supplied as heat to Section A and the piston moves to the right reversibly until the final volume of Section B is \(1.00 \mathrm{dm}^{3}\). Calculate (a) \(\Delta S_{\mathrm{A}}\) and \(\Delta S_{\mathrm{B}}\), (b) \(\Delta A_{\mathrm{A}}\) and \(\Delta \mathrm{A}_{\mathrm{B}}\), (c) \(\Delta G_{\mathrm{A}}\) and \(\Delta G_{\mathrm{B}}\), (d) AS of the total system and its surroundings. If numerical values cannot be obtained, indicate whether the values should be positive, negative, or zero or are indeterminate from the information given. (Assume \(C_{\mathrm{v}, \mathrm{m}}=20 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\).)

Step by step solution

01

- Calculate the Final Temperature of Section A

For section A, since the piston is adiabatic and there is no heat exchange with the surroundings, the process is adiabatic. Using the adiabatic expansion equation for an ideal gas we have: \(PV^\gamma = K\), where \(\gamma = \frac{C_p}{C_v}\). The initial and final states for section A: \(P_1V_1^\gamma = P_2V_2^\gamma\). Since \(C_{v,m}\) is given and we assume the ideal gas has a molar heat capacity at constant pressure, \(C_{p,m}\) which is \(C_{v,m} + R\) (R being the ideal gas constant 8.314 J/mol K), we can find \(\gamma\). \(T_2\) for section A can't be determined with the given information because the pressures are not provided.

02

- Calculate the Change in Entropy for Section A \(\Delta S_A\)

The change in entropy for an adiabatic process is zero because there is no heat exchange with the surroundings, so \(\Delta S_A = 0\).

03

- Calculate the Change in Entropy for Section B \(\Delta S_B\)

For section B, since it's isothermal, the change in entropy \(\Delta S_B\) can be calculated using \(\Delta S = nR\ln\frac{V_f}{V_i}\). With \(n = 2.00 \text{mol}\), \(R = 8.314 \text{J/mol K}\), \(V_i = 2.00 \text{dm}^3\), and \(V_f = 1.00 \text{dm}^3\), we calculate \(\Delta S_B\).

04

- Calculate the Change in Helmholtz Free Energy for Section A \(\Delta A_A\)

The Helmholtz free energy change \(\Delta A\) for an adiabatic process is equal to the work done on or by the system. Since we do not have the pressure or the final temperature for section A, we cannot determine the value or sign of \(\Delta A_A\).

05

- Calculate the Change in Helmholtz Free Energy for Section B \(\Delta A_B\)

For an isothermal process, the change in Helmholtz free energy \(\Delta A\) is given by \(\Delta A = -nRT\ln\frac{V_f}{V_i}\), which can be calculated for section B using the provided values.

06

- Calculate the Change in Gibbs Free Energy for Section A \(\Delta G_A\)

The Gibbs free energy change \(\Delta G_A\) for a process at constant temperature and pressure is given by \(\Delta G_A = \Delta H_A - T\Delta S_A\). However, since this is an adiabatic expansion, \(\Delta G_A\) is not defined as there is no constant temperature process, hence we cannot determine the value or sign of \(\Delta G_A\) with the given information.

07

- Calculate the Change in Gibbs Free Energy for Section B \(\Delta G_B\)

Since the process in section B is isothermal, the change in Gibbs free energy \(\Delta G_B\) is zero for an isothermal process at equilibrium because \(\Delta G = \Delta H - T\Delta S\), and in an isothermal process at constant temperature, \(T\Delta S = \Delta H\). Therefore, \(\Delta G_B = 0\).

08

- Calculate the Change in Entropy for the Total System and Surroundings

The change in entropy for the total system is the sum of the changes of entropy of sections A and B: \(\Delta S_{total} = \Delta S_A + \Delta S_B\). The entire process is reversible, so the change in entropy of the surroundings should be the negative of the change in entropy of the system. Hence, the change in entropy for the surroundings is \(-\Delta S_{total}\), and the combined change in entropy for the system and surroundings is zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adiabatic Expansion

Adiabatic expansion occurs when a gas expands without exchanging heat with its surroundings. In this process, the work done by the gas leads to a change in internal energy, which affects the temperature of the gas. Because there's no heat exchange, the adiabatic process is characterized by the equation
\( PV^\gamma = \text{constant} \),
where \( P \) is the pressure, \( V \) is the volume, and \( \gamma \) (gamma) is the heat capacity ratio, usually more than one for real gases.

Importance in Thermodynamics

In thermodynamics, adiabatic processes are significant because they are examples of energy conservation and provide insight into the behavior of isolated systems. For students, understanding adiabatic processes is essential to mastering concepts in physics and engineering relating to how systems respond to changes without external energy influence.

Isothermal Expansion

Isothermal expansion is a process in which a gas expands while maintaining a constant temperature. This scenario requires heat transfer to or from the surroundings to ensure the gas's temperature remains unchanged. The equation used to define this behavior is
\( PV = \text{constant} \),
and the key relation for calculating entropy change in an isothermal process for an ideal gas is given by:
\( \(Delta S = nRln(\frac{V_f}{V_i}) \) \),
where \( \Delta S \) is the change in entropy, \( n \) is the number of moles of the gas, \( R \) is the ideal gas constant, and \( V_i \) and \( V_f \) are the initial and final volumes, respectively.

Role in Real-World Applications

Isothermal expansions are fundamental to understanding how heat engines and refrigerators operate, as they involve stages where systems exchange heat while keeping their temperature steady.

Entropy Change

Entropy is a measure of the disorder or randomness in a system. The change in entropy, denoted as \( \Delta S \), indicates how much the disorder has changed during a process. In an adiabatic process where no heat is exchanged, the entropy change is zero because there's no increase or decrease in the system's disorder.
On the other hand, for isothermal processes, the entropy change is typically nonzero and is dependent on the volume change as seen with the equation:
\( \Delta S = nRln(\frac{V_f}{V_i}) \).

Understanding Entropy in Thermodynamics

Entropy is a central concept in the second law of thermodynamics. It helps explain why some processes are irreversible and provides insight into the direction of spontaneous processes in isolated systems. The law posits that for an isolated system, entropy can never decrease. Therefore, knowing how to calculate entropy changes aids in predicting the feasibility of chemical and physical changes.

Gibbs Free Energy

Gibbs free energy (\( G \)) is the measure of the maximum reversible work that can be performed by a thermodynamic system at constant temperature and pressure. It is defined by the equation:
\( \Delta G = \Delta H - T\Delta S \),
where \( \Delta G \) represents the change in Gibbs free energy, \( \Delta H \) the change in enthalpy, \( T \) the temperature, and \( \Delta S \) the entropy change. For isothermal processes in ideal gases, if the process is done reversibly, the change in Gibbs free energy can often be zero, indicating an equilibrium state.

Significance of Gibbs Free Energy in Chemistry

In chemical reactions, the sign of \( \Delta G \) determines the spontaneity of a reaction. A negative \( \Delta G \) implies that the reaction proceeds spontaneously, while a positive value suggests non-spontaneity. This concept is pivotal for students in predicting reaction behavior and understanding the relationship between free energy and reaction equilibrium.