If the molar heat capacity of a gas in the process PT= constant is 5R, the number of degrees of freedom of molecules in the gas is

Options

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

Answer

To solve the problem, we need to find the number of degrees of freedom (F) of the gas molecules given that the molar heat capacity (C) in the process where \(PT = \text{constant}\) is \(5R\).

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The process is defined as \(PT = \text{constant}\).
- The molar heat capacity \(C = 5R\).

2. **Relate Molar Heat Capacity to Degrees of Freedom:**
- For a gas, the molar heat capacity at constant volume \(C_V\) is related to the degrees of freedom \(F\) by the equation:
\[ C_V = \frac{F R}{2} \]

3. **Use the Heat Capacity Relation:**
- In the process \(PT = \text{constant}\), we can express the molar heat capacity \(C\) in terms of \(C_V\) and the work done:
\[ C = C_V + P \frac{dV}{dT} \]
- From the ideal gas law \(PV = nRT\), and for one mole of gas (\(n = 1\)), we have:
\[ PV = RT \]
- Thus, we can express \(P\) as:
\[ P = \frac{RT}{V} \]

4. **Differentiate the Volume:**
- To find \(dV\), we need to differentiate the volume \(V\) with respect to temperature \(T\). Since \(PT = k\) (a constant), we can derive the relationship between \(V\) and \(T\).

5. **Substituting Values:**
- After differentiating and substituting, we find:
\[ C = C_V + 2R \]
- Given that \(C = 5R\), we can set up the equation:
\[ 5R = C_V + 2R \]

6. **Solve for \(C_V\):**
- Rearranging gives:
\[ C_V = 5R - 2R = 3R \]

7. **Equate \(C_V\) to Degrees of Freedom:**
- Now, substituting \(C_V\) back into the equation for degrees of freedom:
\[ 3R = \frac{F R}{2} \]
- Cancel \(R\) from both sides:
\[ 3 = \frac{F}{2} \]

8. **Solve for \(F\):**
- Multiplying both sides by 2 gives:
\[ F = 6 \]

### Final Answer:
The number of degrees of freedom of the molecules in the gas is \(F = 6\).