Which of the following paramenters is the same for molecules of all gases at a given temperature?

Options

A. (a)mass
B. (b)Speed
C. (c)Momentum
D. (d)Kinetic energy.

Answer

To solve the question "Which of the following parameters is the same for molecules of all gases at a given temperature?", we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Kinetic Energy**:
The kinetic energy (K.E.) of gas molecules is given by the formula:
\[ K.E. = \frac{1}{2} m v_{\text{rms}}^2 \]
where \( m \) is the mass of the gas molecule and \( v_{\text{rms}} \) is the root mean square velocity of the gas molecules.

2. **Relate Root Mean Square Velocity to Temperature**:
The root mean square velocity \( v_{\text{rms}} \) can be expressed as:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature in Kelvin, and \( M \) is the molar mass of the gas.

3. **Substitute \( v_{\text{rms}} \) into Kinetic Energy Formula**:
By substituting \( v_{\text{rms}} \) into the kinetic energy formula, we get:
\[ K.E. = \frac{1}{2} m \left(\sqrt{\frac{3RT}{M}}\right)^2 \]
This simplifies to:
\[ K.E. = \frac{1}{2} m \cdot \frac{3RT}{M} \]

4. **Simplify the Kinetic Energy Expression**:
Notice that the mass \( m \) cancels out when we express it in terms of the molar mass \( M \):
\[ K.E. = \frac{3RT}{2} \]
Here, \( R \) and \( T \) are constants for a given temperature.

5. **Conclusion**:
The kinetic energy of gas molecules is directly proportional to the temperature and remains the same for all gases at a given temperature. Therefore, the answer to the question is that the **kinetic energy** is the parameter that is the same for molecules of all gases at a given temperature.

### Final Answer:
**Kinetic Energy**

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