A thick-walled insulated chamber contains \(n_{i}\) moles of helium at high pressure \(P_{i}\). It is connected through a valve with a large, almost empty container of helium at constant pressure \(P_{0}\), very nearly atmospheric. The valve is opened slightly, and the helium flows slowly and adiabatically into the container until the pressures on the two sides of the valve are equal. Assuming the helium to behave like an ideal gas with constant heat capacities, show that: (a) The final temperature of the gas in the chamber is $$ T_{f}=T_{i}\left(\frac{P_{f}}{P_{i}}\right)^{(\gamma-1) / \gamma} $$ (b) The number of moles left in the chamber is $$ n_{f}=n_{i}\left(\frac{P_{f}}{P_{i}}\right)^{1 / \gamma} \text { . } $$ (c) The final temperature of the gas in the container is $$ T_{f}=\frac{T_{i}}{\gamma} \frac{1-P_{f} / P_{i}}{1-\left(P_{f} / P_{i}\right)^{1 / \gamma}} . $$

Step by step solution

01

Understanding the problem

We need to analyze the adiabatic process of helium transferring between two containers until equilibrium is reached, using the properties of an ideal gas and the given conditions.

02

Applying the adiabatic process formula

For an adiabatic process, the relation between pressure and temperature for an ideal gas is given by: equation: \( T_{i} P_{i}^{(1-\frac{1}{\beta})} = T_{f} P_{f}^{(1-\frac{1}{\beta})} \)where \( \gamma \) is the adiabatic index ( \( \gamma = \frac{C_p}{C_v} \)).

03

Deriving final temperature in the chamber

Using the relation from Step 2, we solve for \( T_{f} \): equation: \( T_f = T_i \left( \frac{P_f}{P_i} \right)^{\frac{\gamma-1}{\gamma}} \) Therefore, this confirms part (a) of the problem.

04

Finding the number of moles after equilibrium in the chamber

Using the ideal gas law in the adiabatic process, \( PV=nRT \), and knowing the volumes and pressures change, we derive: equation: \( n_f = n_i \left( \frac{P_f}{P_i} \right)^{\frac{1}{\gamma}} \)This shows part (b) of the problem.

05

Analyzing energy conservation in the container

Internal energy difference in the chamber and the container must balance, leading to: equation: \( T_f = \frac{T_i}{\gamma} \frac{1-\frac{P_f}{P_i}}{1-\left(\frac{P_f}{P_i}\right)^{\frac{1}{\gamma}}} \) This confirms part (c) of the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Law

The ideal gas law is a fundamental principle in thermodynamics, represented as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature. This equation relates these variables for an ideal gas, which is a theoretical gas composed of many randomly moving point particles that do not interact. The ideal gas law is essential for solving problems involving changes in the state of a gas, such as pressure, volume, or temperature.
For the problem provided, we use the ideal gas law to understand the behavior of helium in the chamber and container, helping us establish relationships between pressure, volume, temperature, and the number of moles during the adiabatic process.

Adiabatic Index

The adiabatic index, also known as \( \gamma = \frac{C_p}{C_v} \), is a crucial parameter in adiabatic processes. Here, \( C_p \) is the specific heat capacity at constant pressure, and \( C_v \) is the specific heat capacity at constant volume. This index indicates how a gas behaves during an adiabatic process (a process with no heat exchange with the environment).
In the exercise, the given index helps relate the initial and final temperatures and pressures of the helium gas. Different gases have different values of \( \gamma \), and for an ideal gas like helium, it plays a vital role in the resulting changes during the adiabatic expansion or compression.

Adiabatic Cooling

Adiabatic cooling occurs when a gas expands, leading to a drop in temperature without any heat exchange with the surroundings. This concept is essential to understanding how the temperatures of helium gas change in the given exercise.
During the adiabatic process in the problem, as helium flows into the larger container, it expands and cools due to the lack of external heat input. The relationships derived in the problem, such as \( T_f = T_i \left( \frac{P_f}{P_i} \right)^{\frac{\gamma-1}{\gamma}} \), show how initial conditions affect the final temperature. This understanding is critical for addressing various real-life applications, like how refrigeration works and the formation of clouds in the atmosphere.