Assertion : The pressure of a given mass of a gas is directly proportional to the temperature on kelvin scale at constant volume
Reason : With the increase in temperature, the average kinetic energy and hence the average velocity of the molecule increases

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is not a correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the question, we need to analyze both the assertion and the reason provided.

### Step 1: Understand the Assertion
The assertion states that "The pressure of a given mass of a gas is directly proportional to the temperature on the Kelvin scale at constant volume."

- According to the ideal gas law, \( PV = nRT \), where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

Since the volume \( V \), the number of moles \( n \), and the gas constant \( R \) are constant, we can rearrange the equation to show that:

\[ P \propto T \]

This means that if the temperature increases, the pressure will also increase, confirming that the assertion is true.

### Step 2: Understand the Reason
The reason states that "With the increase in temperature, the average kinetic energy and hence the average velocity of the molecule increases."

- The average kinetic energy of a gas is given by the equation:

\[ KE = \frac{3}{2} RT \]

This shows that kinetic energy is directly proportional to temperature. As temperature increases, the average kinetic energy of the gas molecules also increases.

- Additionally, the root mean square velocity (\( v_{rms} \)) of gas molecules can be expressed as:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

Where \( M \) is the molar mass of the gas. This equation also indicates that as temperature increases, the average velocity of the gas molecules increases.

### Step 3: Conclusion
Both the assertion and the reason are true. The reason provided correctly explains the assertion, as the increase in temperature leads to an increase in kinetic energy and velocity, which in turn increases the pressure of the gas when the volume is held constant.

### Final Answer
Both the assertion and reason are true, and the reason is the correct explanation for the assertion.

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