Kinetic Theory of Temperature and Pressure：練習問題

An ideal gas is taken through a cyclic process shown on a \(P-V\) diagram consisting of three steps: an isothermal expansion from state 1 to 2, an isobaric compression from state 2 to 3, and an isochoric pressure increase from state 3 back to 1. If the temperature at state 1 is \(T_1\) and the volume doubles during the isothermal expansion, what is the temperature at state 3?

Two moles of an ideal gas are compressed isothermally from an initial pressure of \(100\text{ kPa}\) to a final pressure of \(400\text{ kPa}\) at a constant temperature of \(300\text{ K}\). What is the work done on the gas during this process?

The figure shows a \(P\text{-}V\) diagram of a cycle for a heat engine. The process \(A \to B\) is isobaric expansion, \(B \to C\) is isochoric cooling, and \(C \to A\) is an isothermal compression. If the pressure at \(A\) is \(P_A = 3.0 \times 10^5\text{ Pa}\) and the volume is \(V_A = 1.0 \times 10^{-3}\text{ m}^3\), and the volume at \(B\) is \(V_B = 3.0 \times 10^{-3}\text{ m}^3\), calculate the work done by the gas during the isobaric expansion \(A \to B\).<\/p>

In a heat engine, an ideal gas undergoes a cycle where it absorbs heat \(Q_H\) from a high-temperature reservoir at \(T_H = 600\text{ K}\) and exhausts heat \(Q_C\) to a low-temperature reservoir at \(T_C = 300\text{ K}\). If the engine performs \(400\text{ J}\) of work per cycle and its efficiency is exactly half of the maximum theoretical (Carnot) efficiency, how much heat is exhausted to the cold reservoir per cycle?<\/p>

An ideal gas with adiabatic index \(γ\) is compressed adiabatically from an initial volume \(V_1\) to a final volume \(V_2 = V_1/8\). If the initial absolute temperature is \(T_1\) and \(γ = 5/3\), what is the final temperature \(T_2\)?

Consider a \( P-V \) diagram of a cyclic process where a gas moves from state A to B (isobaric expansion), B to C (isochoric cooling), and C to A (isothermal compression). Is the net work done by the gas over one full cycle positive, negative, or zero?

A diatomic ideal gas expands adiabatically from volume \( V \) to \( 32V \). If the initial pressure is \( P_0 \), what is the final pressure in terms of \( P_0 \)? Use \( \gamma = 1.4 \).

An ideal gas is contained in a piston-cylinder assembly. If the gas undergoes an adiabatic expansion where the volume doubles, how does the final temperature \( T_f \) compare to the initial temperature \( T_i \)? Assume the ratio of specific heats \( \gamma = 5/3 \).

A heat engine operates between a high-temperature reservoir at \( T_H = 600 \text{ K} \) and a low-temperature reservoir at \( T_L = 300 \text{ K} \). In each cycle, the engine absorbs \( 2000 \text{ J} \) of heat from the hot reservoir and performs \( 800 \text{ J} \) of work.
(a) Calculate the actual thermal efficiency of this engine.
(b) Calculate the maximum possible efficiency (Carnot efficiency) for an engine operating between these two temperatures.
(c) Calculate the total entropy change of the universe per cycle. Is this process reversible or irreversible?

An insulated cylinder contains \( 0.5 \text{ moles} \) of an ideal monatomic gas initially at a pressure of \( 1.0 \times 10^5 \text{ Pa} \) and a volume of \( 0.01 \text{ m}^3 \). The gas is compressed adiabatically until its volume is reduced to \( 0.002 \text{ m}^3 \). For a monatomic gas, \( γ = 5/3 \).
(a) Determine the final pressure of the gas.
(b) Calculate the work done on the gas during this compression.
(c) Explain, in terms of molecular kinetics, why the temperature of the gas increases during this adiabatic compression.