For an ideal gas the fractional change in its volume per degree rise in temperature at constant pressure is equal to [T is absolute temperature of gas]

Options

A. `T^0`
B. `T`
C. `T^(-1)`
D. `T^2`

Answer

To solve the problem of finding the fractional change in volume per degree rise in temperature at constant pressure for an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Start with the Ideal Gas Equation**:
The ideal gas equation is given by:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature.

2. **Differentiate the Ideal Gas Equation**:
Since we are considering a process at constant pressure, we can differentiate both sides of the equation with respect to temperature \( T \):
\[ P \, dV = nR \, dT \]

3. **Rearranging the Equation**:
Rearranging the differentiated equation gives:
\[ dV = \frac{nR}{P} \, dT \]

4. **Expressing the Change in Volume**:
Now, we can express the fractional change in volume \( \frac{dV}{V} \):
\[ \frac{dV}{V} = \frac{nR}{PV} \, dT \]

5. **Using the Ideal Gas Equation to Substitute for \( V \)**:
From the ideal gas equation, we know that \( V = \frac{nRT}{P} \). Substituting this into our equation gives:
\[ \frac{dV}{V} = \frac{nR}{P \cdot \frac{nRT}{P}} \, dT = \frac{1}{T} \, dT \]

6. **Finding the Fractional Change in Volume per Degree Rise in Temperature**:
Now, we can find the fractional change in volume per unit change in temperature:
\[ \frac{dV}{V} \div dT = \frac{1}{T} \]

7. **Final Result**:
Therefore, the fractional change in volume per degree rise in temperature at constant pressure is:
\[ \frac{dV}{V \, dT} = \frac{1}{T} \]

### Conclusion:
The answer is that the fractional change in volume per degree rise in temperature at constant pressure for an ideal gas is equal to \( \frac{1}{T} \).