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 pressure is halved. What are the final volume?

검증된 단계별 안내

Step 1: Understand the problem. This is an adiabatic process, meaning no heat is exchanged with the surroundings. The relationship between pressure, volume, and temperature for an adiabatic process is governed by the equation \( P V^\gamma = \text{constant} \), where \( \gamma \) (gamma) is the adiabatic index, which for diatomic gases like O₂ is approximately 1.4.

Step 2: Write down the given values. The initial pressure \( P_1 = 3.0 \ \text{atm} \), the initial temperature \( T_1 = 150^\circ \text{C} = 423 \ \text{K} \), the number of moles \( n = 0.10 \ \text{mol} \), and the final pressure \( P_2 = 1.5 \ \text{atm} \) (since the pressure is halved).

Step 3: Use the ideal gas law \( PV = nRT \) to calculate the initial volume \( V_1 \). Rearrange the equation to \( V_1 = \frac{nRT_1}{P_1} \), where \( R \) is the ideal gas constant \( 0.0821 \ \text{L·atm·mol}^{-1}\text{·K}^{-1} \). Substitute the known values to find \( V_1 \).

Step 4: Apply the adiabatic condition \( P_1 V_1^\gamma = P_2 V_2^\gamma \) to find the final volume \( V_2 \). Rearrange the equation to \( V_2 = V_1 \left( \frac{P_1}{P_2} \right)^{1/\gamma} \). Substitute \( \gamma = 1.4 \), \( P_1 \), \( P_2 \), and \( V_1 \) to calculate \( V_2 \).

Step 5: Verify the units and ensure consistency throughout the calculations. The final volume \( V_2 \) will be in liters, as the ideal gas constant \( R \) was used in \( \text{L·atm·mol}^{-1}\text{·K}^{-1} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

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 essential for understanding the behavior of gases under various conditions, allowing us to calculate changes in volume or pressure when other variables are altered.

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. Understanding this concept is crucial for analyzing how the gas expands and how its temperature and pressure change during the expansion.

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 helps us understand how the internal energy of the gas changes as it expands, affecting its temperature and pressure while conserving total energy.