Problem 29 - University Physics with Modern Physics

Question

A monatomic ideal gas that is initially at a pressure of \(1.50 \times 10^{5}\) Pa and has a volume of 0.0800 \(\mathrm{m}^{3}\) is compressed adiabatically to a volume of 0.0400 \(\mathrm{m}^{3} .\) (a) What is the final pressure? (b) How much work is done by the gas? (c) What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?

Step-by-Step Solution

Answer

(a) 5.19 × 10⁵ Pa, (b) -10.4 × 10³ J, (c) \(T_2/T_1 = 1.59\); the gas is heated.

Step 1: Understanding the Adiabatic Process

In an adiabatic compression, no heat is exchanged with the environment. We apply the adiabatic condition for an ideal gas: \[ P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \]where \(P_1\) and \(P_2\) are initial and final pressures, \(V_1\) and \(V_2\) are initial and final volumes, and \(\gamma = \frac{C_p}{C_v}\) is the heat capacity ratio. For a monatomic gas, \(\gamma = \frac{5}{3}\).

Step 2: Calculate the Final Pressure

We know \(P_1 = 1.50 \times 10^5\) Pa, \(V_1 = 0.0800\) m³, and \(V_2 = 0.0400\) m³. Using the adiabatic condition:\[ P_2 = P_1 \left( \frac{V_1}{V_2} \right)^{\gamma} \].Substitute the values and \(\gamma = \frac{5}{3}\):\[ P_2 = 1.50 \times 10^5 \times \left( \frac{0.0800}{0.0400} \right)^{\frac{5}{3}} \].Calculate to find \(P_2 = 5.19 \times 10^5\) Pa.

Step 3: Calculate Work Done by the Gas

For an adiabatic process, work done by the gas can be calculated as:\[ W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} \].Substitute \(P_1 = 1.50 \times 10^5\) Pa, \(V_1 = 0.0800\) m³, \(P_2 = 5.19 \times 10^5\) Pa, \(V_2 = 0.0400\) m³, and \(\gamma = \frac{5}{3}\):\[ W = \frac{1.50 \times 10^5 \times 0.0800 - 5.19 \times 10^5 \times 0.0400}{\frac{5}{3} - 1} \].Calculate to find \(W = -10.4 \times 10^3\) J (negative sign indicates work done on the gas).

Step 4: Ratio of Final Temperature to Initial Temperature

For an adiabatic process, the temperature ratio is related by:\[ \frac{T_2}{T_1} = \left( \frac{V_1}{V_2} \right)^{\gamma - 1} \].Substitute \(V_1 = 0.0800\) m³, \(V_2 = 0.0400\) m³, and \(\gamma = \frac{5}{3}\):\[ \frac{T_2}{T_1} = \left( \frac{0.0800}{0.0400} \right)^{\frac{5}{3} - 1} \].Calculate to find \(\frac{T_2}{T_1} = 1.59\).

Step 5: Determine if the Gas is Heated or Cooled

Since the ratio \(\frac{T_2}{T_1} > 1\), it indicates that the final temperature is higher than the initial temperature. Thus, the gas is heated by the compression.

Key Concepts

Ideal Gas

An ideal gas is a theoretical concept used in physics and chemistry to simplify the behavior of gases. It assumes a few key properties which make it easier to predict how gases respond under different conditions:

• Gas particles are small and occupy no volume on their own.
• There are no interactions between gas molecules, except for perfectly elastic collisions.
• Gas behavior adheres to the ideal gas law: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of substance (moles), \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Understanding these assumptions helps predict real-world gases' behavior under certain conditions. This is crucial for solving problems involving temperature, volume, and pressure changes during processes like adiabatic compression.

Adiabatic Compression

Adiabatic compression refers to a process where a gas is compressed without exchanging heat with its surroundings. This means the total heat energy remains constant during compression:

• No heat flow occurs into or out of the gas.
• The gas's internal energy changes solely due to work done on it.

For an ideal gas undergoing adiabatic compression, the relation is given by:\[ P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \]where \( P_1 \) and \( P_2 \) are initial and final pressures, \( V_1 \) and \( V_2 \) are the initial and final volumes, and \( \gamma \) is the heat capacity ratio.
This relation helps calculate the final pressure after compression. When a gas is compressed adiabatically, its temperature increases because the particles collide more frequently and energetically, which elevates their kinetic energy without losing energy to the surroundings.

Heat Capacity Ratio

The heat capacity ratio, also known as the adiabatic index or \( \gamma \), is crucial in determining how gases behave during processes like adiabatic compression. It is defined as the ratio of the molar heat capacities:

• \( C_p \): Heat capacity at constant pressure.
• \( C_v \): Heat capacity at constant volume.

The ratio is given by:\[ \gamma = \frac{C_p}{C_v} \].For a monatomic ideal gas, \( \gamma \) is often \( \frac{5}{3} \).This ratio indicates how much a gas will heat up when compressed without heat loss. Higher values indicate a more significant temperature change for a given volume change during an adiabatic process. This concept is essential in atmospheric physics, engineering, and understanding energy changes in gases under various conditions.