The work done by one mole of ideal gas when its teperature is increased by `1^oC` at constant pressure is

Options

A. `frac{R}{2}`
B. 2R
C. R
D. 4R

Answer

To solve the problem of finding the work done by one mole of an ideal gas when its temperature is increased by \(1^\circ C\) at constant pressure, we can follow these steps:

### Step 1: Understand the relationship between work, pressure, and volume
The work done \(W\) by an ideal gas at constant pressure can be expressed as:
\[ W = -P \Delta V \]
where \(P\) is the pressure and \(\Delta V\) is the change in volume.

### Step 2: Use the ideal gas law
The ideal gas law states:
\[ PV = nRT \]
where:
- \(P\) = pressure,
- \(V\) = volume,
- \(n\) = number of moles,
- \(R\) = universal gas constant,
- \(T\) = temperature in Kelvin.

### Step 3: Determine the change in volume
When the temperature increases by \(1^\circ C\), we can express the change in temperature as:
\[ \Delta T = 1^\circ C = 1 \text{ K} \quad (\text{since } 1^\circ C = 1 \text{ K}) \]
Using the ideal gas law, we can find the change in volume:
\[ \Delta V = \frac{nR \Delta T}{P} \]
Substituting \(n = 1\) mole and \(\Delta T = 1\) K, we get:
\[ \Delta V = \frac{1 \cdot R \cdot 1}{P} = \frac{R}{P} \]

### Step 4: Substitute \(\Delta V\) back into the work formula
Now, substituting \(\Delta V\) into the work equation:
\[ W = -P \Delta V = -P \left(\frac{R}{P}\right) = -R \]

### Step 5: Determine the magnitude of work done
The work done is negative, indicating that work is done on the system. However, if we are interested in the magnitude of the work done, we take the absolute value:
\[ |W| = R \]

### Conclusion
Thus, the work done by one mole of an ideal gas when its temperature is increased by \(1^\circ C\) at constant pressure is:
\[ \text{Work done} = R \]