Four moles of an ideal gas is initially in a state A having pressure `2xx10^(5)N//m^(2)` and temperature 200 K . Keeping pressure constant the gas is taken to state B at temperature of 400K. The gas is then taken to a state C in such a way that its temperature increases and volume decreases. Also from B to C, the magnitude of `(dT)/(dV)` increases. The volume of gas at state C is eaual to its volume at state A. Now gas is taken is initial state A keeping volume constant. A total of 1000 J heat is rejected from the sample in the cyclic process. Take `R=8.3J//K//mol`.
Which graph between temperture T and volume V for the cyclic process is correct.

Options

A.
B.
C.
D.

Answer

To solve the problem step by step, we will analyze the states of the gas and the relationships between pressure, volume, and temperature during the cyclic process.

### Step 1: Identify the initial conditions
- The gas has 4 moles (n = 4).
- Initial pressure (P_A) = \(2 \times 10^5 \, \text{N/m}^2\)
- Initial temperature (T_A) = 200 K

### Step 2: Determine the volume at state A
Using the ideal gas law, \(PV = nRT\):
\[ V_A = \frac{nRT_A}{P_A} \]
Substituting the values:
\[ V_A = \frac{4 \times 8.3 \times 200}{2 \times 10^5} \]
Calculating:
\[ V_A = \frac{6640}{200000} = 0.0332 \, \text{m}^3 \]

### Step 3: Determine the state B conditions
At state B, the pressure remains constant, and the temperature is increased to 400 K. Since pressure is constant:
\[ V_B = \frac{nRT_B}{P_A} \]
Substituting the values:
\[ V_B = \frac{4 \times 8.3 \times 400}{2 \times 10^5} \]
Calculating:
\[ V_B = \frac{13280}{200000} = 0.0664 \, \text{m}^3 \]

### Step 4: Analyze the transition from state B to state C
From state B to state C, the temperature increases while the volume decreases. The volume at state C is equal to the volume at state A (i.e., \(V_C = V_A\)).

### Step 5: Determine the conditions at state C
Since \(V_C = V_A\), we have:
\[ V_C = 0.0332 \, \text{m}^3 \]
We need to find the temperature at state C (T_C). Using the ideal gas law again:
\[ P_B V_C = nRT_C \]
Substituting the values:
\[ 2 \times 10^5 \times 0.0332 = 4 \times 8.3 \times T_C \]
Calculating:
\[ 6666 = 33.2 T_C \implies T_C = \frac{6666}{33.2} \approx 200 \, \text{K} \]

### Step 6: Analyze the final state A conditions
At state A, the gas is kept at constant volume and a total of 1000 J of heat is rejected. The process is cyclic, so the net heat exchange must equal zero.

### Step 7: Determine the graph between T and V
- From A to B: \(T\) increases while \(V\) increases (straight line).
- From B to C: \(T\) increases while \(V\) decreases (curve).
- From C back to A: \(T\) remains constant while \(V\) is constant (horizontal line).

### Conclusion
The correct graph will show:
- An initial increase in temperature with increasing volume from A to B.
- A decrease in volume with an increase in temperature from B to C.
- A constant temperature at state A.