A gas expands against a variable pressure given by `P = (20)/(V)` (where P in atm and V in L). During expansion from volume of 1 litre to 10 litre, the gas undergoes a change in internal energy of 400 J. How much heat is absorbed by the gas during expansion?

Options

A. 46 J
B. 4660 J
C. 5065.8 J
D. 4260 J

Explanation

To solve the problem, we need to determine the heat absorbed by the gas during its expansion. We will use the first law of thermodynamics, which states:

\[ \Delta U = Q + W \]

Where:
- \(\Delta U\) = change in internal energy
- \(Q\) = heat absorbed by the system
- \(W\) = work done by the system

Given:
- Change in internal energy, \(\Delta U = 400 \, \text{J}\)
- Pressure, \(P = \frac{20}{V} \, \text{atm}\)
- Volume changes from \(V_1 = 1 \, \text{L}\) to \(V_2 = 10 \, \text{L}\)

### Step 1: Calculate the work done (W)

The work done by the gas during expansion against a variable pressure is given by:

\[ W = -\int_{V_1}^{V_2} P \, dV \]

Substituting the expression for pressure:

\[ W = -\int_{1}^{10} \frac{20}{V} \, dV \]

### Step 2: Evaluate the integral

The integral of \(\frac{1}{V}\) is \(\ln V\):

\[ W = -20 \int_{1}^{10} \frac{1}{V} \, dV = -20 [\ln V]_{1}^{10} \]

Calculating this gives:

\[ W = -20 (\ln 10 - \ln 1) = -20 \ln 10 \]

Since \(\ln 1 = 0\), we have:

\[ W = -20 \ln 10 \]

### Step 3: Convert the logarithm to a numerical value

Using the approximation \(\ln 10 \approx 2.303\):

\[ W = -20 \times 2.303 = -46.06 \, \text{L atm} \]

### Step 4: Convert work done to joules

To convert from L atm to joules, we use the conversion factor \(1 \, \text{L atm} = 101.325 \, \text{J}\):

\[ W = -46.06 \, \text{L atm} \times 101.325 \, \text{J/L atm} = -4667.02 \, \text{J} \]

### Step 5: Apply the first law of thermodynamics

Now, we can find the heat absorbed by the gas using the first law of thermodynamics:

\[ Q = \Delta U - W \]

Substituting the known values:

\[ Q = 400 \, \text{J} - (-4667.02 \, \text{J}) = 400 + 4667.02 = 5067.02 \, \text{J} \]

### Final Answer

The heat absorbed by the gas during expansion is approximately:

\[ Q \approx 5067.02 \, \text{J} \]