A gas expands slowely against a variable pressure given by `p=(10)/(V)` bar, where V is the volume of gas at each stage of expansion. During expansion from volume 10 L to 100 L the gas undergoes an increase in internal energy of 400 J. How much heat is absorbed by gas during expansion ?

Options

A. 1900 J
B. 2300 J
C. 2700 J
D. 423 J

Answer

To solve the problem, we will use the first law of thermodynamics and the information provided about the gas expansion.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The pressure as a function of volume is given by \( P = \frac{10}{V} \) bar.
- The initial volume \( V_1 = 10 \) L.
- The final volume \( V_2 = 100 \) L.
- The increase in internal energy \( \Delta U = 400 \) J.

2. **Determine the Work Done (W):**
- The work done during an expansion against a variable pressure can be calculated using the formula:
\[ W = -\int_{V_1}^{V_2} P \, dV \]
- Substitute \( P \) with \( \frac{10}{V} \):
\[ W = -\int_{10}^{100} \frac{10}{V} \, dV \]
- This integral evaluates to:
\[ W = -10 \ln \left( \frac{V_2}{V_1} \right) = -10 \ln \left( \frac{100}{10} \right) = -10 \ln(10) \]
- Calculate \( W \):
\[ W = -10 \times 2.302 = -23.02 \, \text{J} \quad (\text{approximately}) \]

3. **Apply the First Law of Thermodynamics:**
- The first law of thermodynamics states:
\[ \Delta U = Q + W \]
- Rearranging gives:
\[ Q = \Delta U - W \]
- Substitute the values:
\[ Q = 400 \, \text{J} - (-23.02 \, \text{J}) = 400 \, \text{J} + 23.02 \, \text{J} = 423.02 \, \text{J} \]

4. **Conclusion:**
- The heat absorbed by the gas during the expansion is approximately \( Q = 423.02 \) J.

### Final Answer:
The heat absorbed by the gas during expansion is approximately **423.02 J**.