The energy of a given sample of an ideal gas depends only on its

Options

A. (a)volume
B. (b)pressure
C. (c) density
D. (d)temperature

Answer

To solve the question regarding the energy of a given sample of an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Kinetic Energy in Gases**:
The kinetic energy (K) of an ideal gas is related to the motion of its molecules. For an ideal gas, the kinetic energy can be expressed using the formula:
\[ K = \frac{1}{2} M v_{\text{rms}}^2 \]
where \( M \) is the molecular mass of the gas and \( v_{\text{rms}} \) is the root mean square velocity of the gas molecules.

2. **Finding the Expression for Root Mean Square Velocity**:
The root mean square velocity (\( v_{\text{rms}} \)) is given by the formula:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the gas constant and \( T \) is the absolute temperature in Kelvin.

3. **Substituting \( v_{\text{rms}} \) into the Kinetic Energy Formula**:
We can substitute the expression for \( v_{\text{rms}} \) into the kinetic energy formula:
\[ K = \frac{1}{2} M \left(\sqrt{\frac{3RT}{M}}\right)^2 \]
Simplifying this, we have:
\[ K = \frac{1}{2} M \cdot \frac{3RT}{M} \]
The \( M \) cancels out:
\[ K = \frac{3}{2} RT \]

4. **Identifying the Dependence of Kinetic Energy**:
From the final expression \( K = \frac{3}{2} RT \), we can see that the kinetic energy \( K \) is directly proportional to the temperature \( T \). This means that the energy of a given sample of an ideal gas depends only on its temperature.

5. **Conclusion**:
Therefore, the correct answer to the question is that the energy of a given sample of an ideal gas depends only on its **temperature**.

### Final Answer:
The energy of a given sample of an ideal gas depends only on its **temperature**.

---