One mole of an ideal gas is contained with in a cylinder by a frictionless piston and is initially at temperature T. The pressure of the gas is kept constant while it is heated and its volume doubles. If R is molar gas constant, the work done by the gas in increasing its volume is

Options

A. RT ln2
B. 1/2 RT
C. RT
D. 3/2 RT

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Given Information
We have one mole of an ideal gas, initially at temperature \( T \). The pressure of the gas is kept constant while it is heated, and its volume doubles.

### Step 2: Use the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = molar gas constant
- \( T \) = temperature

Since we have one mole of gas (\( n = 1 \)), we can rewrite the equation as:
\[ PV = RT \]

### Step 3: Initial and Final Volumes
Let the initial volume of the gas be \( V \). When the gas is heated, its volume doubles, so the final volume \( V_f \) is:
\[ V_f = 2V \]

### Step 4: Work Done by the Gas
The work done by the gas during an isobaric process (constant pressure) is given by:
\[ W = P \Delta V \]
where \( \Delta V \) is the change in volume. The change in volume is:
\[ \Delta V = V_f - V = 2V - V = V \]

### Step 5: Substitute into the Work Done Equation
Now, we can substitute \( \Delta V \) into the work done equation:
\[ W = P \cdot V \]

### Step 6: Use the Ideal Gas Law to Find Pressure
From the ideal gas law, we know:
\[ PV = RT \implies P = \frac{RT}{V} \]

### Step 7: Substitute Pressure into Work Done Equation
Now substitute \( P \) back into the work done equation:
\[ W = \left(\frac{RT}{V}\right) \cdot V = RT \]

### Conclusion
Thus, the work done by the gas in increasing its volume is:
\[ \boxed{RT} \]

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