Consider a container like that shown in the Figure, with n1n_1n1 moles of a monatomic gas on one side and n2n_2n2 moles of a diatomic gas on the other. The monatomic gas has initial temperature T1iT_{1i}T1i. The diatomic gas has initial temperature T2iT_{2i}T2i. Show that the equilibrium temperature isTf=3n1T1i+5n2T2i3n1+5n2T_f = \frac{3n_1 T_{1i} + 5n_2 T_{2i}}{3n_1 + 5n_2}

Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

문제 73b

20장, 문제 73b

Consider a container like that shown in the Figure, with $n_1$ moles of a monatomic gas on one side and $n_2$ moles of a diatomic gas on the other. The monatomic gas has initial temperature $T_{1i}$ . The diatomic gas has initial temperature $T_{2i}$ . Show that the equilibrium temperature is

$T_{f} = \frac{3 n_{1} T_{1 i} + 5 n_{2} T_{2 i}}{3 n_{1} + 5 n_{2}}$

검증된 단계별 안내

Step 1: Understand the problem. We are tasked with finding the equilibrium temperature when two gases, one monatomic and one diatomic, are allowed to exchange heat in a thermally insulated container. The system is isolated, so the total internal energy remains constant.

Step 2: Write the expression for the internal energy of each gas. For a monatomic gas, the internal energy is given by $ U_1 = \frac{3}{2} n_1 R T_1 $, where $ n_1 $ is the number of moles, $ R $ is the gas constant, and $ T_1 $ is the temperature. For a diatomic gas, the internal energy is $ U_2 = \frac{5}{2} n_2 R T_2 $, where $ n_2 $ is the number of moles and $ T_2 $ is the temperature.

Step 3: Apply the principle of conservation of energy. At equilibrium, the total internal energy of the system remains constant. This means the sum of the initial internal energies of the two gases equals the sum of their internal energies at equilibrium: $ \frac{3}{2} n_1 R T_{1i} + \frac{5}{2} n_2 R T_{2i} = \frac{3}{2} n_1 R T_{eq} + \frac{5}{2} n_2 R T_{eq} $.

Step 4: Simplify the equation to solve for the equilibrium temperature $ T_{eq} $. Factor out $ R $ and rearrange terms: $ T_{eq} = \frac{3 n_1 T_{1i} + 5 n_2 T_{2i}}{3 n_1 + 5 n_2} $.

Step 5: Interpret the result. The equilibrium temperature $ T_{eq} $ is a weighted average of the initial temperatures $ T_{1i} $ and $ T_{2i} $, with the weights determined by the number of moles and the degrees of freedom of each gas. This result reflects the distribution of energy according to the equipartition theorem.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Thermodynamic Equilibrium

Thermodynamic equilibrium occurs when a system's macroscopic properties, such as temperature and pressure, become uniform throughout. In the context of gases, this means that after sufficient time, the temperatures of the monatomic and diatomic gases will equalize, leading to a single equilibrium temperature. This concept is crucial for understanding how energy is distributed between different types of gases in a closed system.

Heat Capacity

Heat capacity is the amount of heat required to change a substance's temperature by a given amount. For gases, the heat capacity can vary depending on whether the gas is monatomic or diatomic. Monatomic gases typically have a lower heat capacity than diatomic gases due to their simpler molecular structure, which affects how they store thermal energy. This concept is essential for calculating the final equilibrium temperature when two gases with different heat capacities are mixed.

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the two gases, the heat lost by the hotter gas will equal the heat gained by the cooler gas, leading to the establishment of an equilibrium temperature. This principle is fundamental for analyzing energy exchanges in thermodynamic processes and is key to solving problems involving multiple gases.