Assertion : Different gases at the same conditions of temperature and pressure have same root mean square speed
Reason : Average K.E. of a gas is directly proportional to temperature in kelvin

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 "Different gases at the same conditions of temperature and pressure have the same root mean square speed."

- The root mean square speed (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature in Kelvin, and \( M \) is the molar mass of the gas.

### Step 2: Analyze the Assertion
- From the formula, we can see that the root mean square speed depends on both the temperature (T) and the molar mass (M) of the gas.
- If we have different gases (which means they have different molar masses), even at the same temperature, their root mean square speeds will not be the same because they have different values of \( M \).
- Therefore, the assertion is **false**.

### Step 3: Understand the Reason
The reason states that "Average K.E. of a gas is directly proportional to temperature in Kelvin."

- The average kinetic energy (K.E.) of a gas is given by:
\[ K.E. = \frac{3}{2}RT \]
- This formula shows that the average kinetic energy is indeed directly proportional to the temperature (T) in Kelvin.

### Step 4: Analyze the Reason
- Since the average kinetic energy increases with an increase in temperature, the reason is **true**.

### Conclusion
- The assertion is false, and the reason is true. Thus, the correct answer is that the assertion is false and the reason is true.

### Final Answer
- Assertion: False
- Reason: True