Problem 160 - A Complete Resource Book in Chemistry for JEE Main

Question

Two closed bulbs of equal volume \((V)\) containing an ideal gas initially at pressure \(P\) and temperature \(T_{1}\) are connected through a narrow tube of negligible volume as shown in the figure below. The temperature of one of the bulbs is then raised to \(T_{2}\). The final pressure \(p_{f}\) is [2016] (a) \(2 p_{i}\left(\frac{T_{1}}{T_{1}+T_{2}}\right)\) (b) \(2 p_{i}\left(\frac{T_{2}}{T_{1}+T_{2}}\right)\) (c) \(2 p_{i}\left(\frac{T_{1} T_{2}}{T_{1}+T_{2}}\right)\) (d) \(p_{i}\left(\frac{T_{1} T_{2}}{T_{1}+T_{2}}\right)\)

Step-by-Step Solution

Answer

The final pressure is \(2 p_i \left( \frac{T_1 T_2}{T_1 + T_2} \right) \), answer (c).

Step 1: Initial Conditions Analysis

There are two bulbs each with a volume \( V \) containing the ideal gas at pressure \( P \) and temperature \( T_1 \). Initially, both bulbs have the same temperature \( T_1 \) and the total volume of gas is \( 2V \) because each bulb is of volume \( V \).

Step 2: Ideal Gas Law for Initial State

Using the ideal gas law \( PV = nRT \), the initial amount of moles in each bulb can be expressed as \( n_1 = \frac{PV}{RT_1} \). Therefore, the total initial moles of gas in both bulbs combine to give \( n_{total} = 2n_1 = \frac{2PV}{RT_1} \).

Step 3: Final Conditions After Heating

One bulb is heated to a temperature \( T_2 \), while the other remains at \( T_1 \). Let's assume the moles in the bulb at \( T_2 \) is \( n_2 \) and at \( T_1 \) is \( n_1 \). In the final state, consistent with the conservation of moles, \( n_{total} = n_1 + n_2 \).

Step 4: Application of Ideal Gas Law for Each Bulb in Final State

Using the ideal gas law for each bulb, we get \( p_f V = n_1 R T_1 \) and \( p_f V = n_2 R T_2 \). This implies \( n_1 = \frac{p_f V}{RT_1} \) and \( n_2 = \frac{p_f V}{RT_2} \).

Step 5: Solving for Final Pressure \( p_f \)

Substitute \( n_1 \) and \( n_2 \) from the final state equation into the relation from the conservation of moles: \[ \frac{2PV}{RT_1} = \frac{p_f V}{RT_1} + \frac{p_f V}{RT_2} \]. Simplifying gives \[ 2P = p_f \left( \frac{T_1 + T_2}{T_1T_2} \right) \]. Hence, \( p_f = 2P \left(\frac{T_1T_2}{T_1 + T_2}\right) \).

Step 6: Final Answer Selection

Comparing the expression for \( p_f \) with the given options, the correct answer is (c): \( 2 p_i \left( \frac{T_1 T_2}{T_1 + T_2} \right) \).

Key Concepts

Pressure in Gases

Pressure is a fundamental concept in physics and chemistry, especially when dealing with gases. It is defined as the force exerted by gas molecules on the walls of their container per unit area. In simpler terms, it tells us how much the gas molecules are pushing against the walls of the container.

When examining the pressure in a gas, it's essential to consider the factors that can influence it:

• Number of molecules: More molecules mean more collisions, leading to higher pressure.
• Temperature: Increasing the temperature gives the molecules more energy, causing them to move faster and increase pressure.
• Volume of the container: A smaller volume means less space for molecules to move around, resulting in more frequent collisions and higher pressure.

In the ideal gas law, pressure is directly linked to temperature and the number of gas molecules, which is why understanding pressure is crucial in predicting how gases will behave under different conditions.

Temperature and Its Role

Temperature is a measure of the average kinetic energy of the particles in a substance. For gases, this means temperature directly affects how fast the molecules are moving.

In the context of the ideal gas law:

• Higher temperatures mean faster-moving molecules. This can lead to an increase in pressure if the gas is confined.
• Changing the temperature of a gas can impact other properties like volume and pressure, if other variables are kept constant.

In scenarios like our exercise, where one bulb is heated to a higher temperature, this results in a different behavior of gas on either side of the connected bulbs. This temperature difference is a driving factor for changes in pressure and distribution of gas.

Understanding Moles of Gas

Moles are a way to express the amount of a substance. In the context of gases, one mole refers to a specific number of gas molecules, approximately \(6.022 \times 10^{23}\) molecules, known as Avogadro's number.

The ideal gas law ties the concept of moles to pressure, volume, and temperature:

• The number of moles directly correlates with the amount of gas present in a container.
• More moles of gas mean more molecules, which can increase pressure if within the same volume.

In our exercise, the calculations involve determining how the distribution of moles changes given a temperature shift in the bulbs, while maintaining the total number of moles constant.

Conservation of Moles

Conservation of moles is a vital concept when analyzing closed systems involving gases. It states that the total number of gas moles remains constant within a closed system, assuming no gas can enter or escape the system.

In the ideal gas law context, this principle helps us track how gas behaves when conditions such as temperature or pressure change. When applied to our exercise:

• The total moles in the system stayed constant, despite the temperature change influencing the distribution.
• This principle ensures that while individual bulb conditions change, the overall quantity of gas does not.

By knowing that the moles are conserved, we were able to solve for the final pressure, understanding how gas behavior balanced out across the bulbs.