A fixed mass of helium gas is made to undergo a process in which its pressure varies linearly from `1 kPa` to `2 kPa`, in relation to its volume as the latter varies from `0.2 m^(3)` to `0.4 m^(3)`. The heat absorbed by the gas will be

Options

A. `300 J`
B. `900 J`
C. `1200 J`
D. `1500 J`

Explanation

To solve the problem of calculating the heat absorbed by a fixed mass of helium gas undergoing a linear pressure-volume process, we can follow these steps:

### Step 1: Understand the Process
We have a fixed mass of helium gas with pressure varying from \( P_1 = 1 \, \text{kPa} \) to \( P_2 = 2 \, \text{kPa} \) and volume varying from \( V_1 = 0.2 \, \text{m}^3 \) to \( V_2 = 0.4 \, \text{m}^3 \).

### Step 2: Draw the PV Diagram
Plot the points on a Pressure-Volume (PV) diagram. The initial point (1) is at (0.2 m³, 1 kPa) and the final point (2) is at (0.4 m³, 2 kPa). The line connecting these points represents the process.

### Step 3: Calculate Change in Internal Energy
Using the formula for change in internal energy for an ideal gas:
\[ \Delta U = n C_v \Delta T \]
We can also express it in terms of pressure and volume:
\[ \Delta U = \frac{P_2 V_2 - P_1 V_1}{\gamma - 1} \]
Where \( \gamma = \frac{C_p}{C_v} \) for helium (a monatomic gas) is \( \frac{5}{3} \).

Substituting the values:
\[ \Delta U = \frac{(2 \, \text{kPa} \times 0.4 \, \text{m}^3) - (1 \, \text{kPa} \times 0.2 \, \text{m}^3)}{\frac{5}{3} - 1} \]
Calculating:
\[ \Delta U = \frac{(0.8 - 0.2) \, \text{kPa m}^3}{\frac{2}{3}} = \frac{0.6 \, \text{kPa m}^3}{\frac{2}{3}} = 0.9 \, \text{kJ} = 900 \, \text{J} \]

### Step 4: Calculate Work Done
The work done by the gas during this process can be calculated as the area under the PV curve, which forms a trapezium:
\[ W = \frac{1}{2} \times (P_1 + P_2) \times (V_2 - V_1) \]
Substituting the values:
\[ W = \frac{1}{2} \times (1 \, \text{kPa} + 2 \, \text{kPa}) \times (0.4 \, \text{m}^3 - 0.2 \, \text{m}^3) \]
Calculating:
\[ W = \frac{1}{2} \times 3 \, \text{kPa} \times 0.2 \, \text{m}^3 = 0.3 \, \text{kJ} = 300 \, \text{J} \]

### Step 5: Apply the First Law of Thermodynamics
According to the first law of thermodynamics:
\[ Q = \Delta U + W \]
Substituting the values we found:
\[ Q = 900 \, \text{J} + 300 \, \text{J} = 1200 \, \text{J} \]

### Conclusion
The heat absorbed by the gas is:
\[ \boxed{1200 \, \text{J}} \]