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Ch 19: Work, Heat, and the First Law of Thermodynamics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Knight Calc 5th EditionCh 19: Work, Heat, and the First Law of Thermodynamics문제 32a
19장, 문제 32a

A gas cylinder holds 0.10 mol of O₂ at 150°C and a pressure of 3.0 atm. The gas expands adiabatically until the volume is doubled. What are the final pressure?

검증된 단계별 안내
1
Step 1: Understand the problem. This is an adiabatic process, meaning no heat is exchanged with the surroundings. For an adiabatic process involving an ideal gas, the relationship between pressure, volume, and temperature is governed by the adiabatic condition: \( P V^\gamma = \text{constant} \), where \( \gamma \) (gamma) is the adiabatic index, defined as \( \gamma = \frac{C_p}{C_v} \). For diatomic gases like O₂, \( \gamma \approx 1.4 \).
Step 2: Write the adiabatic condition for the initial and final states. Let the initial pressure, volume, and temperature be \( P_1 \), \( V_1 \), and \( T_1 \), and the final pressure, volume, and temperature be \( P_2 \), \( V_2 \), and \( T_2 \). Since \( V_2 = 2V_1 \), the adiabatic condition becomes: \( P_1 V_1^\gamma = P_2 V_2^\gamma \). Substitute \( V_2 = 2V_1 \) into the equation.
Step 3: Solve for the final pressure \( P_2 \). Rearrange the adiabatic condition to isolate \( P_2 \): \( P_2 = P_1 \left( \frac{V_1}{V_2} \right)^\gamma \). Substitute \( V_2 = 2V_1 \) and simplify: \( P_2 = P_1 \left( \frac{1}{2} \right)^\gamma \). Use \( \gamma = 1.4 \) for diatomic gases.
Step 4: Use the ideal gas law to relate the initial conditions. The ideal gas law is \( PV = nRT \). For the initial state, \( P_1 V_1 = nRT_1 \). Convert the given temperature from Celsius to Kelvin: \( T_1 = 150 + 273 = 423 \, \text{K} \). This relationship can help verify the initial volume if needed.
Step 5: Recognize that the final temperature \( T_2 \) can also be determined using the adiabatic relation between temperature and volume: \( T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1} \). Rearrange to solve for \( T_2 \): \( T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{\gamma-1} \). Substitute \( V_2 = 2V_1 \) and simplify: \( T_2 = T_1 \left( \frac{1}{2} \right)^{\gamma-1} \).

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주요 개념

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

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is fundamental in understanding the behavior of gases under various conditions, allowing us to calculate changes in state variables when the gas undergoes transformations.
추천 영상:
가이드 코스
07:21
Ideal Gases and the Ideal Gas Law

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. For an ideal gas, this means that any change in internal energy is due solely to work done on or by the gas. In such processes, the relationship between pressure and volume can be described by the adiabatic equation, which is crucial for determining the final state of the gas after expansion.
추천 영상:
가이드 코스
06:13
Entropy & Ideal Gas Processes

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. In the context of an adiabatic process, this law implies that the work done by the gas during expansion results in a change in internal energy, affecting its temperature and pressure. Understanding this principle is essential for analyzing energy changes in thermodynamic systems.
추천 영상:
가이드 코스
08:04
The First Law of Thermodynamics
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