Problem 20 - Arihant AIEEE Physics

Question

A gas is contained in a closed vessel at \(250 \mathrm{~K}\), then the percentage increase in pressure, if the gas is heated through \(1^{\circ} \mathrm{C}\), is : (a) \(0.4 \%\) (b) \(0.6 \%\) (c) \(0.8 \%\) (d) \(1.0 \%\)

Step-by-Step Solution

Answer

The percentage increase in pressure is 0.4%, which corresponds to option (a).

Step 1: Understanding the Problem

We need to calculate the percentage increase in pressure when the temperature of a gas in a closed vessel is increased by 1°C. The initial temperature is 250 K.

Step 2: Identify Relevant Formula

From the ideal gas law, we know that pressure is directly proportional to temperature when volume and the amount of gas remain constant. We can use the relation \( P_1/T_1 = P_2/T_2 \).

Step 3: Define Initial and Final Temperatures

The initial temperature \( T_1 = 250 \) K. An increase of \(1^{\circ} \mathrm{C} \) corresponds to an increase of \(1 \) K, so \( T_2 = 251 \) K.

Step 4: Calculate Initial and Final Pressures

Since \( P \propto T \), we can write: \[ \frac{P_2}{P_1} = \frac{T_2}{T_1} = \frac{251}{250} \]

Step 5: Calculate Percentage Increase in Pressure

The percentage increase in pressure is given by \[ \left(\frac{P_2 - P_1}{P_1}\right) \times 100\% = \left(\frac{251/250 - 1}{1}\right) \times 100\% \] Simplifying gives \[ \frac{1}{250} \times 100\% = 0.4\% \]

Step 6: Select the Correct Answer

The percentage increase in pressure is \(0.4\%\). Thus, the correct answer is option (a).

Key Concepts

Pressure-Temperature Relationship

The pressure-temperature relationship is a fundamental concept stemming from the ideal gas law. The ideal gas law tells us that pressure (\( P \)) is directly related to temperature (\( T \)) when volume and the amount of gas are constant. This principle is crucial in understanding how gases behave under different conditions. The formula representing this relationship is:\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]Where:

• \( P_1 \) is the initial pressure.
• \( P_2 \) is the final pressure.
• \( T_1 \) is the initial temperature.
• \( T_2 \) is the final temperature.

This equation shows a proportionate change. If temperature increases, the pressure will also increase provided the volume and the number of moles of gas stay the same. This relationship helps predict how changing the temperature affects the pressure in scenarios like heating a gas in a closed container. Understanding this can help prevent accidents by predicting pressure increases that could lead to a burst container.

Temperature Conversion

Temperature conversion between Celsius and Kelvin is a simple yet important process in thermodynamics. This conversion is necessary because many scientific formulas, including the ideal gas law, require temperature in Kelvin. Here’s a quick guide:

• To convert Celsius to Kelvin, simply add 273.15.
• For example, \( 1^{\circ} \mathrm{C} \) is equivalent to \( 274.15 \mathrm{~K} \).
• Conversely, to convert Kelvin to Celsius, you subtract 273.15.

In the given exercise, the increase of \(1^{\circ} \mathrm{C} \) directly translates to an increase of \(1 \mathrm{~K} \). This straightforward conversion is important for calculations, ensuring that the units are consistent and accurate conclusions can be drawn from the mathematical relations involving gas laws.

Percentage Change Calculation

Calculating percentage change is an essential arithmetic skill used across various fields, especially when analyzing changes in physical conditions like pressure and temperature. In this exercise, we determined how the pressure changes as temperature increases slightly.The percentage change is calculated using the formula:\[ \left(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}\right) \times 100\% \]Applying this concept to pressure changes:

• The original value translates to initial pressure \(P_1\).
• The new value corresponds to the final pressure \(P_2\) after temperature change.

In this exercise, the pressure change is derived from the ratio \(\frac{P_2}{P_1} = \frac{251}{250}\), where the increase in pressure results from the change in temperature by \(1 \mathrm{~K} \). By simplifying, we found the percentage increase in pressure to be \(0.4\%\), highlighting how even a small rise in temperature significantly affects gas pressure.