`N_2` is confined in a cylindrical vessel with a movable piston exposed to open atmosphere. If 25 kcal of heat is added to it and the internal energy of the gas increases by 8 kcal, find the work done by the gas.

Answer

To solve the problem, we will apply 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 given values:**
- Heat added to the gas, \(\Delta Q = 25 \text{ kcal}\)
- Change in internal energy, \(\Delta U = 8 \text{ kcal}\)

2. **Apply the first law of thermodynamics:**
According to the first law, we can rearrange the equation to find the work done by the gas:
\[ W = \Delta Q - \Delta U \]

3. **Substitute the known values into the equation:**
\[ W = 25 \text{ kcal} - 8 \text{ kcal} \]

4. **Calculate the work done:**
\[ W = 17 \text{ kcal} \]

Thus, the work done by the gas is **17 kcal**.