To find the heat given to the gas during an isobaric process, we can use the relationship between work done, heat, and the degrees of freedom of the gas.
### Step-by-Step Solution:
1. **Identify the Given Values**:
- Work done by the gas, \( W = 10 \, \text{J} \)
- Degrees of freedom, \( f = 5 \)
2. **Understand the Isobaric Process**:
- In an isobaric process, the pressure remains constant. The relationship between heat added to the system (\( Q \)), work done by the system (\( W \)), and the change in internal energy (\( \Delta U \)) is given by:
\[
Q = \Delta U + W
\]
3. **Relate Internal Energy Change to Degrees of Freedom**:
- For an ideal gas, the change in internal energy (\( \Delta U \)) can be expressed in terms of the degrees of freedom:
\[
\Delta U = \frac{f}{2} n R \Delta T
\]
- However, we can also relate \( Q \) directly to \( W \) using the specific heat capacity at constant pressure, which is related to degrees of freedom.
4. **Calculate the Heat Transfer**:
- The relationship between heat transfer and work done in an isobaric process is given by:
\[
Q = \frac{f + 2}{f} W
\]
- Substituting the values:
\[
Q = \frac{5 + 2}{5} \times 10 \, \text{J}
\]
\[
Q = \frac{7}{5} \times 10 \, \text{J} = 14 \, \text{J}
\]
5. **Final Calculation**:
- Since we need to find the heat given to the gas, we can also use the formula:
\[
Q = \frac{7}{2} W
\]
- Substituting \( W = 10 \, \text{J} \):
\[
Q = \frac{7}{2} \times 10 = 35 \, \text{J}
\]
### Conclusion:
The heat given to the gas during the isobaric process is \( Q = 35 \, \text{J} \).
