A polyatomic gas with six degrees of freedom does 25J of work when it is expanded at constant pressure. The heat given to the gas is

Options

A. 100J
B. 150J
C. 200J
D. 250J

Answer

To solve the problem, we need to determine the heat given to the polyatomic gas when it expands at constant pressure and does 25 J of work. We will use the first law of thermodynamics, which states:

\[ \Delta Q = \Delta U + W \]

Where:
- \(\Delta Q\) is the heat added to the system,
- \(\Delta U\) is the change in internal energy,
- \(W\) is the work done by the system.

### Step-by-Step Solution:

1. **Identify the Degrees of Freedom**:
The problem states that the gas has 6 degrees of freedom, which means \(f = 6\).

2. **Calculate the Heat Capacity at Constant Volume (\(C_v\))**:
The heat capacity at constant volume for a polyatomic gas is given by:
\[ C_v = \frac{f}{2} R = \frac{6}{2} R = 3R \]

3. **Relate Work Done to Change in Volume**:
The work done by the gas during expansion at constant pressure is given as:
\[ W = P \Delta V = 25 \, \text{J} \]

4. **Calculate Change in Internal Energy (\(\Delta U\))**:
The change in internal energy is given by:
\[ \Delta U = n C_v \Delta T \]
To find \(\Delta T\), we can use the relationship between work done and change in volume. However, we can also relate the heat capacity at constant pressure \(C_p\) to \(C_v\):
\[ C_p = C_v + R = 3R + R = 4R \]

Since we are expanding at constant pressure, we can use the formula:
\[ W = n R \Delta T \]
Rearranging gives:
\[ \Delta T = \frac{W}{nR} = \frac{25 \, \text{J}}{nR} \]

5. **Substituting \(\Delta T\) into \(\Delta U\)**:
Now substituting \(\Delta T\) into the equation for \(\Delta U\):
\[ \Delta U = n C_v \Delta T = n (3R) \left(\frac{25}{nR}\right) = 75 \, \text{J} \]

6. **Applying the First Law of Thermodynamics**:
Now we can substitute \(\Delta U\) and \(W\) into the first law equation:
\[ \Delta Q = \Delta U + W = 75 \, \text{J} + 25 \, \text{J} = 100 \, \text{J} \]

### Final Answer:
The heat given to the gas is \( \Delta Q = 100 \, \text{J} \).

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