A gas has molar heat capacity `C = 4.5 R` in the process `PT = constant`. The number of degrees of freedom of molecules in the gas is

Options

A. `4`
B. `3`
C. `6`
D. `5`

Explanation

To determine the number of degrees of freedom of molecules in a gas with a given molar heat capacity \( C = 4.5 R \) during a process where \( PT = \text{constant} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
We have the molar heat capacity \( C = 4.5 R \) for the process \( PT = \text{constant} \). We need to find the degrees of freedom \( f \) of the gas molecules.

2. **Use the Relationship Between Heat Capacities**:
For an ideal gas undergoing a process where \( PT \) is constant, we can use the first law of thermodynamics:
\[ dq = du + dw \]
where \( dq \) is the heat added, \( du \) is the change in internal energy, and \( dw \) is the work done.

3. **Express Heat Capacity in Terms of Internal Energy**:
The molar heat capacity at constant pressure \( C \) can be expressed as:
\[ C = C_v + dw \]
where \( C_v \) is the molar heat capacity at constant volume.

4. **Calculate the Work Done**:
For the process \( PT = \text{constant} \), we can derive the work done \( dw \) as follows:
\[ dw = P dV \]
Using the ideal gas law, we can express \( P \) in terms of \( T \) and \( V \).

5. **Relate \( C \) and \( C_v \)**:
From the earlier steps, we can express \( C \) in terms of \( C_v \):
\[ C = C_v + 2R \]
Substituting \( C = 4.5R \):
\[ 4.5R = C_v + 2R \]
Rearranging gives:
\[ C_v = 4.5R - 2R = 2.5R \]

6. **Relate \( C_v \) to Degrees of Freedom**:
The molar heat capacity at constant volume is related to the degrees of freedom \( f \) by the equation:
\[ C_v = \frac{f}{2} R \]
Setting this equal to our expression for \( C_v \):
\[ 2.5R = \frac{f}{2} R \]

7. **Solve for Degrees of Freedom \( f \)**:
Dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ 2.5 = \frac{f}{2} \]
Multiplying both sides by 2:
\[ f = 5 \]

8. **Conclusion**:
The number of degrees of freedom of the molecules in the gas is \( f = 5 \).

### Final Answer:
The number of degrees of freedom of molecules in the gas is **5**.

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