An ideal gas on heating from 100 K to 109 K shows an increase by `a %` in its volume at constant P. The value of a is _____________.

Explanation

To solve the problem, we will use the ideal gas law and the concept of volume change with temperature at constant pressure. Here is the step-by-step solution:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Temperature in Kelvin

### Step 2: Identify Constants
Since the problem states that the pressure \( P \) and the number of moles \( n \) are constant, we can deduce that the ratio \( \frac{V}{T} \) is also constant.

### Step 3: Set Up the Volume-Temperature Relationship
From the ideal gas law, we can write:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
Where:
- \( V_1 \) = Initial volume
- \( V_2 \) = Final volume
- \( T_1 = 100 \, K \) (initial temperature)
- \( T_2 = 109 \, K \) (final temperature)

### Step 4: Calculate the Volume Ratio
Using the relationship established:
\[ \frac{V_2}{V_1} = \frac{T_2}{T_1} \]
Substituting the values:
\[ \frac{V_2}{V_1} = \frac{109}{100} = 1.09 \]

### Step 5: Express the Volume Change
The increase in volume can be expressed as:
\[ V_2 - V_1 = V_2 - V_1 = (1.09 V_1 - V_1) = 0.09 V_1 \]

### Step 6: Calculate Percentage Increase in Volume
The percentage increase in volume is given by:
\[ \text{Percentage Increase} = \frac{V_2 - V_1}{V_1} \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \frac{0.09 V_1}{V_1} \times 100 = 9 \]

### Conclusion
Thus, the value of \( a \) is:
\[ a = 9 \]

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