Problem 196 - Master Resource Book in JEE Main Physics

Question

Two thermally insulated vessels 1 and 2 are filled with air at temperatures \(\left(T_{1}, T_{2}\right)\) volumes \(\left(V_{1}, V_{2}\right)\) and pressures ( \(p_{1}, p_{2}\) ) respectively of the value joining the two vessels is opened the temperature inside the vessel at equilibrium will be \(\quad\) [AIEEE 2008, 04] (a) \(T_{1}+T_{2}\) (b) \(\left(T_{1}+T_{2}\right) / 2\) (c) \(\frac{T_{1} T_{2}\left(p_{1} V_{1}+p_{2} V_{2}\right)}{p_{1} V_{1} T_{2}+p_{2} V_{2} T_{1}}\) (d) \(\frac{T_{1} T_{2}\left(p_{1} V_{1}+p_{2} V_{2}\right)}{p_{1} V_{1} T_{1}+p_{2} V_{2} T_{2}}\)

Step-by-Step Solution

Answer

The equilibrium temperature is given by option (c).

Step 1: Understanding the Problem

There are two insulated vessels with different temperatures, volumes, and pressures. When they are connected, they will reach a thermal equilibrium, and we need to find the equilibrium temperature.

Step 2: Using Ideal Gas Law

For an ideal gas, we have \( pV = nRT \). Here, \( p, V \) are pressure and volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature. We use this to express the moles for each vessel: \( n_1 = \frac{p_1 V_1}{RT_1} \) and \( n_2 = \frac{p_2 V_2}{RT_2} \).

Step 3: Expressing Total Moles After Connection

Once the vessels are connected, the total moles of gas is the sum of moles from both vessels: \( n_{total} = n_1 + n_2 = \frac{p_1 V_1}{RT_1} + \frac{p_2 V_2}{RT_2} \).

Step 4: Using Internal Energy Conservation

Since the vessels are thermally insulated, the energy is conserved. The internal energy \( U \) is proportional to the temperature for an ideal gas (monatomic gas internal energy \( U = \frac{3}{2}nRT \)). So, the total energy is \( U_1 + U_2 = U_f \).

Step 5: Setting Up Equation for Equilibrium Temperature

Now use \( U = \frac{3}{2}nRT \):\[ \frac{3}{2} \left( n_1T_1 + n_2T_2 \right) = \frac{3}{2} \times n_{total}T_f \].The factor \( \frac{3}{2}R \) cancels out, leading to:\[ n_1T_1 + n_2T_2 = \left( \frac{p_1 V_1}{RT_1} + \frac{p_2 V_2}{RT_2} \right) T_f \].

Step 6: Solve for Equilibrium Temperature

Rearrange the equation to solve for \( T_f \):\[ T_f = \frac{T_1 \cdot \frac{p_1 V_1}{T_1} + T_2 \cdot \frac{p_2 V_2}{T_2}}{\frac{p_1 V_1}{T_1} + \frac{p_2 V_2}{T_2}} \]Simplifying, we arrive at:\[ T_f = \frac{T_1 T_2(p_1V_1 + p_2V_2)}{p_1V_1T_2 + p_2V_2T_1} \].

Step 7: Matching Solution with Given Options

This calculated expression matches option (c). Therefore, the equilibrium temperature is:\[ T_f = \frac{T_1 T_2(p_1 V_1 + p_2 V_2)}{p_1 V_1 T_2 + p_2 V_2 T_1} \] which corresponds to answer (c).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle used to describe the behavior of gases. It combines several important gas laws into one equation: \[ p V = n R T \] Where:

• \( p \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant.
• \( T \) is the temperature in Kelvin.

By making assumptions that particles do not interact and occupy negligible space, we arrive at this simplified equation, which is used to investigate various properties like pressure, volume, and temperature of gases. In this problem, the Ideal Gas Law helps us calculate the number of moles in each of the thermally insulated vessels.

Internal Energy Conservation

Internal energy conservation is key when dealing with thermally insulated systems. This principle tells us that if no energy enters or leaves a system, the total internal energy remains the same. For gases, the internal energy is closely linked to temperature. When gases intermingle without exchanging energy with the outside, the total internal energies of the systems before and after mixing remain equivalent. The internal energy for an ideal gas can be calculated using the expression: \[ U = \frac{3}{2} n R T \] Here, \( U \) is the internal energy. After the vessels are connected, the energy conservation equation essentially ensures that: \[ U_1 + U_2 = U_f \] Where \( U_1 \) and \( U_2 \) are the initial energies of each vessel, and \( U_f \) is the final energy at equilibrium. This concept allows us to relate the variables we have to find the equilibrium temperature.

Thermally Insulated Vessels

Thermally insulated vessels help demonstrate principles like energy conservation and equilibrium. Such vessels do not exchange heat with their surroundings, meaning any processes occurring inside them are adiabatic. This insulation allows for simplified scenarios in physical problems since we assume no energy loss to the environment. In our exercise, these vessels are crucial because they ensure that energy interactions are strictly between the gases in the vessels themselves. The lack of external influences means that the internal energy principles can be applied straightforwardly. It clearly sets the stage for focusing solely on the behavior and reactions between the two systems (the vessels) once they are joined.

Moles Calculation

Calculating the number of moles in the system is essential to understanding gas behaviors and interactions in the vessels. The Ideal Gas Law can be rearranged to find the moles of gas in each vessel: \[ n = \frac{p V}{R T} \] This formula shows us that the number of moles depends directly on the pressure and volume of the gas, and inversely on both the universal gas constant \( R \) and the temperature \( T \). For each vessel before connection, the moles are calculated as follows: - For the first vessel: \[ n_1 = \frac{p_1 V_1}{R T_1} \]- For the second vessel: \[ n_2 = \frac{p_2 V_2}{R T_2} \] By determining \( n_1 \) and \( n_2 \), we can understand the contribution of each vessel to the final equilibrium state.

AIEEE 2008 Physics Problem

This exercise is derived from a previous AIEEE (All India Engineering Entrance Examination) paper from 2008, which tests students on core physics principles like gas laws and energy conservation. The focus in this problem is about finding the equilibrium temperature when two ideal gases mix in insulated vessels. Physics problems such as this are great practice for demonstrating an understanding of theoretical principles and applying them in practical, tangible contexts. Key skills tested include:

• Identifying relevant principles for a given scenario.
• Applying mathematical calculations to real-world situations.
• Deriving complex solutions from simpler equations.

Solving such problems requires a grasp of physics fundamentals and critical thinking. It's not merely about reaching an answer but about understanding the journey of getting there.