The equation of state for a gas is given by `PV = eta RT + alpha V`, where `eta` is the number of moles and `alpha` a positive constant. The intinal pressure and temperature of 1 mol of the gas contained in a cylinder is `P_(0)` and `T_(0)`, respectively. The work done by the gas when its temperature doubles isobarically will be

Options

A. `(P_(0) T_(0) R)/(P_(0) - alpha)`
B. `(P_(0) T_(0) R)/(P_(0) + alpha)`
C. `P_(0) T_(0) R` 1n 2
D. none of these

Explanation

To solve the problem, we need to find the work done by the gas during an isobaric process when its temperature doubles. Let's break down the solution step by step.

### Step 1: Understand the Work Done in an Isobaric Process
The work done (W) by a gas during an isobaric process is given by the formula:
\[ W = P \Delta V \]
where \( P \) is the constant pressure and \( \Delta V \) is the change in volume.

### Step 2: Use the Given Equation of State
The equation of state for the gas is given by:
\[ PV = \eta RT + \alpha V \]
where \( \eta \) is the number of moles, \( R \) is the universal gas constant, \( \alpha \) is a positive constant, and \( V \) is the volume.

### Step 3: Rearranging the Equation
We can rearrange the equation to express pressure \( P \):
\[ P = \frac{\eta RT + \alpha V}{V} \]
This can be simplified to:
\[ P = \frac{\eta R}{V} T + \alpha \]

### Step 4: Determine the Change in Volume
When the temperature doubles (from \( T_0 \) to \( 2T_0 \)), we need to find the change in volume \( \Delta V \). The change in temperature \( \Delta T \) is:
\[ \Delta T = 2T_0 - T_0 = T_0 \]

### Step 5: Express \( \Delta V \) in Terms of \( P \) and \( \alpha \)
From the equation of state, we can express the change in volume:
\[ P \Delta V = \eta R \Delta T + \alpha \Delta V \]
Rearranging gives:
\[ P \Delta V - \alpha \Delta V = \eta R \Delta T \]
Factoring out \( \Delta V \):
\[ \Delta V (P - \alpha) = \eta R \Delta T \]
Thus,
\[ \Delta V = \frac{\eta R \Delta T}{P - \alpha} \]

### Step 6: Substitute Values
Substituting \( \Delta T = T_0 \) and \( \eta = 1 \) (since we have 1 mole of gas):
\[ \Delta V = \frac{R T_0}{P - \alpha} \]

### Step 7: Substitute \( \Delta V \) into Work Done Formula
Now substituting \( \Delta V \) back into the work done formula:
\[ W = P \Delta V = P \left(\frac{R T_0}{P - \alpha}\right) \]
This simplifies to:
\[ W = \frac{P R T_0}{P - \alpha} \]

### Final Answer
Thus, the work done by the gas when its temperature doubles is:
\[ W = \frac{P_0 R T_0}{P_0 - \alpha} \]