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Problems on Thermodynamics and Entropy of Ideal Gases

## Problem 1: Second Law of Thermodynamics and Entropy of Ideal Gases

Let the specific heat ratio be γ and the gas constant be R. Here, is the constant volume specific heat and is the constant pressure specific heat. If there is nothing to say about the ideal gas, it means the ideal gas with the amount of substance dealt with in class of 1 mol. For each problem, write not only the answer but also the derivation. Points may be deducted if the answer is only the answer or if the derivation is insufficient.

Answer the following questions about the second law of thermodynamics and the entropy of ideal gases.

(1) Briefly explain the contents of the second law of thermodynamics. Various expressions areknown in the second law of thermodynamics, but only one answer is required (10 points).

(2) Calculate and obtain the temperature T / volume V dependence S (T, V) of the entropy of 1 mol of the ideal gas (10 points).Hint: Equation dU = TdS − PdV that combines the first and second laws of thermodynamics in an infinitesimal quasi-static process Should be integrated.

(3) An ideal gas with a temperature of T and 1 mol is enclosed in a container A having a volume of. This container A is connected to the container B via a valve. At first, the valve was closed and the inside of container B was evacuated. Also, these containers are surrounded by a heat insulating material. When the valve was opened, adiabatic free expansion was observed in which the ideal gas spread to the container on the vacuum side, and the volume of the ideal gas became \$ in the final state. Find the change in entropy in this process (10 points).

The Carnot cycle is a type of two-temperature heat engine, where the temperature of the high-temperature heat source is and the temperature of the low-temperature heat source. The cycle is realized by connecting four states, and the four states are named A, B, C, D

(volumes are VA, VB, VC, VD, respectively). The Carnot cycle consists of four quasi-states: process 1 (A → B): isothermal expansion, process 2 (B → C): adiabatic expansion, process 3 (C → D): isothermal compression, process 4 (D → A): adiabatic compression. It is realized by connecting static processes. Hereinafter, it is referred to as a working substance.

Consider 1 mol of ideal gas. Answer the following questions.

(1) Draw a P − V diagram of the Carnot cycle and state what the area of the enclosed area means (5 points).

(2) Draw a T − S diagram of the Carnot cycle and state what the area of the enclosed area means (5 points).

(3) Explain the definition of thermal efficiency of a two-temperature heat engine (5 points).

(4) Calculate the thermal efficiency of the Carnot cycle and organize it into a form that uses only the temperature of the heat source. (15 points).

(5) Calculate the thermal efficiency of the Carnot cycle when a photon gas is used as the working substance, and organize it into a form that uses only the temperature of the heat source (5 points).

Since the equation dU = TdS − PdV, which is a combination of the first law and the second law of thermodynamics in an infinitesimal quasi-static process, is in the form of total derivative, various relational expressions can be derived.

(1) Derive the following relational expression (10 points).

(2) Called Gibbs free energy. It was written that dU = TdS − PdV for internal energy, but find a similar equation for the derivative of G (10 points).

(3) Starting from dU = T dS − P dV, derive the energy equation shown below (10 points).

(4) The state equation of van der Waals gas, which is a kind of non-ideal gas, is written

In the ideal gas, TVγ-1was constant in the quasi-static adiabatic process, but in the van der Waals gas, it is the same.(5 points).