A gas undergoes a process such that `pprop1/T`. If the molar heat capacity for this process is `C=33.24J//mol-K`, find the degree of freedom of the molecules of the gas.

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the relationship between pressure and temperature
We are given that pressure \( P \) is inversely proportional to temperature \( T \). This can be expressed mathematically as:
\[ P \propto \frac{1}{T} \quad \Rightarrow \quad P \cdot T = \text{constant} \]

### Step 2: Relate the process to the ideal gas law
Using the ideal gas law, we can express temperature \( T \) in terms of pressure \( P \) and volume \( V \):
\[ PV = nRT \quad \Rightarrow \quad T = \frac{PV}{nR} \]
Substituting this into our earlier relationship gives:
\[ P \cdot \frac{PV}{nR} = \text{constant} \]
This simplifies to:
\[ P^2V = \text{constant} \]

### Step 3: Express the relationship in terms of a power
From \( P^2V = \text{constant} \), we can express this as:
\[ PV^{1/2} = \text{constant} \]
This indicates that \( x = \frac{1}{2} \) in the general form \( PV^x = \text{constant} \).

### Step 4: Use the formula for molar heat capacity
The molar heat capacity \( C \) for a process described by \( PV^x = \text{constant} \) is given by:
\[ C = C_V + \frac{R}{1 - x} \]
Where \( C_V \) is the molar heat capacity at constant volume and \( R \) is the ideal gas constant.

### Step 5: Substitute the known values
We know \( C = 33.24 \, \text{J/mol-K} \) and \( R = 8.31 \, \text{J/mol-K} \). Substituting \( x = \frac{1}{2} \) into the equation gives:
\[ 33.24 = C_V + \frac{8.31}{1 - \frac{1}{2}} \]
This simplifies to:
\[ 33.24 = C_V + \frac{8.31}{\frac{1}{2}} = C_V + 16.62 \]
Thus, we can solve for \( C_V \):
\[ C_V = 33.24 - 16.62 = 16.62 \, \text{J/mol-K} \]

### Step 6: Relate \( C_V \) to the degree of freedom
The relationship between the heat capacity at constant volume and the degree of freedom \( f \) is given by:
\[ C_V = \frac{f}{2} R \]
Substituting \( C_V = 16.62 \) and \( R = 8.31 \):
\[ 16.62 = \frac{f}{2} \cdot 8.31 \]
Solving for \( f \):
\[ f = \frac{16.62 \cdot 2}{8.31} = 4 \]

### Final Answer
The degree of freedom of the molecules of the gas is \( f = 4 \).

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