Problem 91 - Master Resource Book in JEE Main Physics

Question

When a gas filled in a closed vessel is heated through \(1^{\circ} \mathrm{C}\), its pressure increases by \(0.4 \%\). The initial temperature of the gas was (a) \(250 \mathrm{~K}\) (b) \(2500 \mathrm{~K}\) (c) \(250^{\circ} \mathrm{C}\) (d) \(25^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The initial temperature of the gas was 250 K.

Step 1: Understand the problem

The problem is about the relationship between temperature and pressure in a gas that's kept in a closed container. When heated by 1°C, the gas pressure increases by 0.4%. This is a due to one of the gas laws.

Step 2: Apply Gay-Lussac's Law

Gay-Lussac's Law states that pressure is directly proportional to temperature when volume is kept constant, represented as \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). Here, \(P_2 = P_1(1 + 0.004)\) since pressure increases by 0.4%. We need to find \(T_1\), the initial temperature.

Step 3: Set up the equation

With Gay-Lussac's formula, substitute: \( \frac{P}{T} = \frac{P(1+0.004)}{T+1} \). Simplify to find the initial temperature.

Step 4: Solve for Initial Temperature \(T\)

Rearranging the equation gives \( T(1 + 0.004) = T+1 \). Solving gives: \( 0.004T = 1 \) thus, \( T = \frac{1}{0.004} = 250 \).

Step 5: Interpret the result

The initial temperature was found to be 250 K based on the calculations. This corresponds to option (a).

Key Concepts

Temperature-Pressure Relationship

Understanding the temperature-pressure relationship in gases is crucial for grasping Gay-Lussac's Law. When a gas is contained within a closed vessel, its pressure tends to change if the temperature changes.
This occurs because molecules move more quickly and collide with the walls of the vessel more frequently, resulting in higher pressure. In this exercise, we learned that a 1°C rise in temperature causes a 0.4% increase in pressure. This indicates how temperature can substantially impact the pressure of a gas. Some key points about this relationship include:

• Temperature and pressure are directly proportional.
• As the temperature rises, the kinetic energy of gas molecules increases.
• Faster moving molecules lead to increased collisions within the vessel, raising the pressure.

By understanding these points, you can better anticipate changes in gas behavior under different temperature conditions.

Gas Laws

Gas laws are fundamental concepts that describe how gases behave under various conditions. One of these essential laws is Gay-Lussac's Law.Gay-Lussac's Law specifically examines the relationship between temperature and pressure when the volume is kept constant. The equation representing this law is:\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]It states that the ratio of the initial pressure to initial temperature is equal to the ratio of the final pressure to final temperature.
This relationship allows us to predict how pressure changes with temperature in a controlled environment.
**Key takeaways from gas laws include:**

• They assume ideal conditions where gases behave predictably.
• They help calculate changes in gas properties under different scenarios.
• They inform practical applications, such as cooking or inflating tires.

Mastering these gas laws enables a deeper understanding of the physical properties and behavior of gases.

Initial Temperature Calculation

Calculating the initial temperature of a gas using Gay-Lussac's Law can be a straightforward process with practice.In this scenario, we used the given increase in pressure to determine the starting temperature. By applying the law's formula:\[ \frac{P}{T} = \frac{P(1+0.004)}{T+1} \]We simplified to derive:\[ T(1 + 0.004) = T+1 \]To solve for the initial temperature \( T \), follow these steps:

• Rearrange the formula to emphasize \( T \).
• Calculate the change: \( 0.004T = 1 \).
• Find \( T \) by \( T = \frac{1}{0.004} = 250 \).

This calculation shows us the initial temperature was 250 K, following through with the steps provided. Understanding each aspect of the calculation is vital for accurately predicting gas behavior in similar situations.