Two moles of a certain gas at a temperature `T_0=300K` were cooled isochorically so that the pressure of the gas got reduced 2 times. Then as a result of isobaric process, the gas is allowed to expand till its temperature got back to the initial value. Find the total amount of heat absorbed by gas in this process.

Explanation

To solve the problem, we will go through the steps of the isochoric and isobaric processes, calculating the heat absorbed by the gas in each step.

### Step 1: Understand the initial conditions
We have 2 moles of a gas at an initial temperature \( T_0 = 300 \, K \).

### Step 2: Isochoric Process
In the isochoric process, the volume remains constant, and the pressure of the gas is reduced to half. According to the ideal gas law, since the volume is constant, the pressure is directly proportional to the temperature. Therefore, if the pressure is halved, the temperature will also be halved.

\[ T_f = \frac{T_0}{2} = \frac{300 \, K}{2} = 150 \, K \]

### Step 3: Calculate Heat for Isochoric Process
The heat absorbed or released during an isochoric process can be calculated using the formula:

\[ Q_1 = n C_v \Delta T \]

Where:
- \( n = 2 \, \text{moles} \)
- \( C_v \) is the molar heat capacity at constant volume
- \( \Delta T = T_f - T_0 = 150 \, K - 300 \, K = -150 \, K \)

Substituting the values:

\[ Q_1 = 2 \cdot C_v \cdot (-150) = -300 C_v \]

### Step 4: Isobaric Process
In the isobaric process, the gas expands at constant pressure until it returns to its initial temperature of \( 300 \, K \). The heat absorbed during this process is given by:

\[ Q_2 = n C_p \Delta T \]

Where:
- \( \Delta T = T_f - T_i = 300 \, K - 150 \, K = 150 \, K \)

Substituting the values:

\[ Q_2 = 2 \cdot C_p \cdot 150 = 300 C_p \]

### Step 5: Total Heat Absorbed
The total heat absorbed by the gas during the entire process is the sum of the heat absorbed in both processes:

\[ Q = Q_1 + Q_2 \]

Substituting the expressions we found:

\[ Q = (-300 C_v) + (300 C_p) = 300 (C_p - C_v) \]

Using the relation \( C_p - C_v = R \) (where \( R \) is the gas constant), we can rewrite this as:

\[ Q = 300 R \]

### Step 6: Calculate the Total Heat Absorbed
Given that \( R = 8.31 \, J/(mol \cdot K) \):

\[ Q = 300 \cdot 8.31 = 2493 \, J \]

### Final Answer
The total amount of heat absorbed by the gas in this process is \( 2493 \, J \).

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