At constant pressure, the ratio of increases in volume of an ideal gas per degree rise in kelvin temperature to its volume is

Options

A. (a)`T^2`
B. (b)`1/T`
C. (c)`T^3`
D. (d)T

Answer

To solve the problem, we need to find the ratio of the increase in volume of an ideal gas per degree rise in Kelvin temperature to its volume at constant pressure.

### Step-by-Step Solution:

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

2. **Express Volume in Terms of Temperature**:
Rearranging the ideal gas law, we can express volume \( V \) as:
\[ V = \frac{nRT}{P} \]
Since \( n \), \( R \), and \( P \) are constants for a given gas at constant pressure, we can say that volume \( V \) is directly proportional to temperature \( T \):
\[ V \propto T \]

3. **Establish the Relationship Between Volume and Temperature**:
From the proportionality, we can write:
\[ \frac{V_1}{V_2} = \frac{T_1}{T_2} \]
where \( V_1 \) and \( V_2 \) are the volumes at temperatures \( T_1 \) and \( T_2 \) respectively.

4. **Calculate the Change in Volume**:
If we consider a small change in temperature, \( \Delta T = T_2 - T_1 \), we can express the change in volume as:
\[ \Delta V = V_2 - V_1 \]
Rearranging gives:
\[ \Delta V = V_1 \left( \frac{T_2}{T_1} - 1 \right) \]

5. **Substituting for Temperature Change**:
If we take \( \Delta T = 1 \) degree (which is the change in temperature), we can express \( T_2 \) as:
\[ T_2 = T_1 + 1 \]
Therefore, the change in volume can be expressed as:
\[ \Delta V = V_1 \left( \frac{T_1 + 1}{T_1} - 1 \right) = V_1 \left( \frac{1}{T_1} \right) \]

6. **Finding the Ratio**:
Now, we need to find the ratio of the increase in volume per degree rise in temperature to the initial volume:
\[ \text{Ratio} = \frac{\Delta V}{V_1} = \frac{V_1 \left( \frac{1}{T_1} \right)}{V_1} = \frac{1}{T_1} \]

### Final Answer:
The ratio of the increase in volume of an ideal gas per degree rise in Kelvin temperature to its volume is:
\[ \frac{1}{T} \]