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, we need to determine the fractional change in volume per degree rise in temperature for an ideal gas at constant pressure. We will use the ideal gas law and calculus to derive the required relationship.

### Step-by-Step Solution:

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

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

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

4. **Expressing Fractional Change in Volume**:
The fractional change in volume \( \frac{dV}{V} \) can be expressed as:
\[ \frac{dV}{V} = \frac{dV}{dT} \cdot \frac{dT}{V} \]

5. **Substituting for \( \frac{dV}{dT} \)**:
From the previous step, we can substitute \( \frac{dV}{dT} \):
\[ \frac{dV}{V} = \frac{nR}{P} \cdot \frac{dT}{V} \]

6. **Using the Ideal Gas Law to Substitute for \( V \)**:
From the ideal gas law, we can express \( V \) as:
\[ V = \frac{nRT}{P} \]
Substituting this into our equation gives:
\[ \frac{dV}{V} = \frac{nR}{P} \cdot \frac{dT}{\frac{nRT}{P}} = \frac{dT}{T} \]

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

8. **Final Result**:
Thus, we find that the fractional change in volume per degree rise in temperature at constant pressure is:
\[ \frac{dV}{V} \div dT = T^{-1} \]

### Conclusion:
The answer is \( T^{-1} \), which corresponds to option 3.