Problem 4 - Arihant AIEEE Physics

Question

Two chambers, one containing ' \(m_{1}^{\prime} \mathrm{g}\) of a gas at ' \(P_{1}\) ' pressure and other containing ' \(m_{2} \mathrm{~g}\) of a gas at \({ }^{\prime} P_{2}{\underline{\phantom{xx}}}^{\prime}\) pressure, are put in communication with each other. If temperature remains constant, the common pressure reached will be : (a) \(\frac{P_{1} P_{2}\left(m_{1}+m_{2}\right)}{P_{2} m_{1}+P_{1} m_{2}}\) (b) \(\frac{m_{1} m_{2}\left(P_{1}+P_{2}\right)}{\left(P_{2} m_{1}+P_{1} m_{2}\right)}\) (c) \(\frac{P_{1} P_{2} m_{1}}{P_{2} m_{1}+P_{1} m_{2}}\) (d) \(\frac{m_{1} m_{2} P_{2}}{\left(P_{2} m_{1}+m_{2} P_{1}\right)}\)

Step-by-Step Solution

Answer

The common pressure reached is given by option (b).

Step 1: Understand the System

We have two chambers with gases at different masses and pressures: Chamber 1 with mass \(m_1\) and pressure \(P_1\), and Chamber 2 with mass \(m_2\) and pressure \(P_2\). When the chambers are connected, the gases mix and reach a common pressure.

Step 2: Assumptions and Variables Definition

Assume ideal gas behavior, meaning the number of moles is proportional to mass at constant temperature. Let \(P_f\) be the final common pressure of the gas after mixing under constant temperature conditions.

Step 3: Application of Ideal Gas Laws

Applying the ideal gas law \(PV = nRT\), from each chamber initially: - Chamber 1: \(P_1V_1 = n_1RT\) - Chamber 2: \(P_2V_2 = n_2RT\). Where \(n_1\) and \(n_2\) represent moles.

Step 4: Calculate Total Moles After Mixing

Post mixing, the final pressure \(P_f\) applies to total volume \(V_1 + V_2\). So, \(P_f(V_1 + V_2) = (n_1 + n_2)RT\). By substitution of total number of moles: \[ P_f = \frac{n_1P_1V_1 + n_2P_2V_2}{V_1 + V_2}. \]

Step 5: Substitute to Find the Common Pressure

Since \(n_1 = \frac{m_1}{M}\) and \(n_2 = \frac{m_2}{M}\) where \(M\) is molar mass and constant for both chambers with same gas type, - Substitute these in the common pressure equation: \[ P_f = \frac{m_1P_1 + m_2P_2}{m_1 + m_2}. \]

Step 6: Determine the Correct Choice

Compare our derived formula \(P_f = \frac{m_1P_1 + m_2P_2}{m_1 + m_2}\) with options given: - The equation matches with option \(b\), considering each specific term setup.

Key Concepts

Pressure Mixing

When we talk about pressure mixing in a system of connected gas chambers, we are referring to how two different gases, each at their own initial pressures, combine to reach a common equilibrium pressure. Initially, each chamber contains gas at distinct pressures. Upon removing the barrier between chambers, the gases mix freely, redistributing their molecules until the pressure throughout the system is uniform.

• **Equilibrium**: The gas molecules diffuse across the chambers until both reach the same pressure.
• **Final Pressure**: This common pressure, or final pressure, is influenced by the initial pressures of the individual chambers and the masses of gas present.

The mixing process follows the principles of the ideal gas law which stipulates that gases mix in such a way that the total pressure is a function of their individual pressures and the number of moles they contribute to the system.

Gas Chambers

Gas chambers in this context are containers that house gases under certain conditions of pressure and volume. The problem scenario usually involves connecting these chambers and observing the behavior of the gas particles within them. This arrangement allows for studying how gases interact under similar conditions when their container size changes instantly due to intercommunication.

• **Initial Conditions**: Each chamber has its own specific mass and pressure of the gas it contains.
• **Communication**: When chambers are connected, barriers are removed, allowing for gas exchange and balancing of pressures.

Each chamber can be seen as a mini-environment where gases behave according to the ideal gas law. The introduction to another chamber mixes the gases, resulting in the calculation of a common pressure based on initial conditions.

Temperature Constant

In our scenario, we assume that the temperature remains constant during the mixing of the gases. This is a critical consideration as it simplifies calculations by eliminating variables associated with temperature change.

• **Impact of Constant Temperature**: When the temperature is held constant, the kinetic energy of gas molecules is unchanged, which simplifies our application of the ideal gas law.
• **Ideal Gas Law**: At a constant temperature, the ideal gas law is expressed as \( PV = nRT \), where the volume and the number of moles (\(n\)) become primary considerations in solving the equation for pressure.

This constancy ensures that the derived equation for the common pressure directly relates masses and pressures without worrying about thermal expansion or contraction.

Moles Calculation

The calculation of moles is crucial in applying the ideal gas law and understanding the behavior of the gas mixtures. Given the assumption of ideal gases, the number of moles is directly related to the mass and molar mass of the gas.

• **Mole Definition**: A mole is a unit that measures the amount of substance, linked to the atomic or molecular mass of the gas.
• **Proportionality**: In our calculations, the moles of gas are proportional to the mass of the gas, given by \( n = \frac{m}{M} \) where \(m\) is the mass and \(M\) is the molar mass.

By substituting the moles equation into the pressure equation, we are able to solve for the common pressure: \( P_f = \frac{m_1P_1 + m_2P_2}{m_1 + m_2} \). This showcases how the initially provided quantities of gas masses can ultimately determine equilibrium pressure through their mole calculation.