Problem 90 - 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 is 250 K (option a).

Step 1: Understand the problem

We need to determine the initial temperature of a gas in a closed vessel when its pressure increases by 0.4% upon heating through 1°C.

Step 2: Use the Ideal Gas Law

For a gas, the relation between pressure and temperature is given by the equation \( \frac{P_2}{P_1} = \frac{T_2}{T_1} \), where \(P_1\) and \(P_2\) are initial and final pressures, and \(T_1\) and \(T_2\) are initial and final temperatures respectively.

Step 3: Set up the equation using percentage change

Since the pressure increases by 0.4%, \( P_2 = P_1 + 0.004 P_1 = 1.004 P_1 \). The temperature increase of 1°C implies \( T_2 = T_1 + 1 \). Substitute these into the Ideal Gas Law equation.

Step 4: Solve for initial temperature

Substituting into the equation gives \( \frac{1.004P_1}{P_1} = \frac{T_1 + 1}{T_1} \), simplifying which gives \( 1.004 = \frac{T_1 + 1}{T_1} \). Solving this, we have \( 1.004T_1 = T_1 + 1 \), which simplifies to \( 0.004T_1 = 1 \), yielding \( T_1 = 250 \).

Step 5: Determine the correct option

The initial temperature in Kelvin is found to be 250 K. The corresponding option is (a) 250 K.

Key Concepts

Pressure-Temperature Relationship

In the context of gases, the pressure-temperature relationship is crucial to understand how a gas's behavior changes with temperature. According to Gay-Lussac's Law, for a given amount of gas at constant volume, the pressure of the gas is directly proportional to its temperature in Kelvin. This means when the temperature increases, the pressure does too, and vice versa. The mathematical expression of this relationship is given by:

• \( \frac{P_2}{P_1} = \frac{T_2}{T_1} \)

Where \( P_1 \) and \( P_2 \) are the initial and final pressures, while \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin respectively. This proportion indicates a linear relationship, so doubling the temperature will double the pressure, as long as the volume and amount of gas remain constant. This foundational principle is part of the Ideal Gas Law, which encompasses the behaviors of gases under various conditions.

Thermal Expansion of Gases

Thermal expansion refers to the way in which matter, including gases, changes in volume in response to changes in temperature. When a gas is heated, its particles move more energetically, causing an increase in pressure if the gas is confined to a constant volume. This is because the particles collide with the walls of the container more frequently and with greater force.In closed systems, such as a sealed vessel, we primarily observe changes in pressure rather than volume. The Ideal Gas Law, represented as \( PV = nRT \), helps explain this behavior, suggesting that temperature changes significantly affect gas pressure in such cases. This concept is vital for understanding how gases behave under temperature changes in all sorts of practical applications, from industrial processes to everyday weather phenomena.

Percentage Change in Pressure

Understanding percentage changes in pressure can help illustrate how sensitive gases are to changes in temperature. In this context, a 0.4% increase means that the final pressure is 0.4% greater than the initial pressure.To calculate this, we adjust the final pressure using:

• \( P_2 = P_1 + 0.004P_1 = 1.004P_1 \)

This simple calculation shows the precision needed when working with gases, where even small changes in temperature can lead to noticeable pressure differences. It underscores the importance of accurate pressure measurements, especially in scientific experiments or industrial applications where control over gas conditions is essential.

Calculation of Initial Temperature

Determining the initial temperature of a gas can help predict how it will behave under additional heating or cooling. In the provided problem, determining the initial temperature followed from using the given percentage pressure increase and the corresponding temperature increase.The solution involves rearranging and solving the equation derived from the Ideal Gas Law. Starting with the expression:

• \( \frac{1.004P_1}{P_1} = \frac{T_1 + 1}{T_1} \)

We simplify to:

• \( 1.004 = \frac{T_1 + 1}{T_1} \)
• This leads to: \( 1.004T_1 = T_1 + 1 \)
• Solving for \( T_1 \), we find \( 0.004T_1 = 1 \), hence \( T_1 = 250 \) Kelvin

This calculation highlights the mathematical precision needed in thermodynamics and the utility of formulas like the Ideal Gas Law in deriving important gas properties. Understanding these relations is crucial for students and professionals dealing with real-world gas systems.