If an ideal gas at 100 K is heated to 109 K in a rigid container, the pressure increases by X%. What is the value of X?

Explanation

To solve the problem, we will use the relationship between pressure and temperature for an ideal gas in a rigid container. Here’s the step-by-step solution:

### Step 1: Understand the relationship between pressure and temperature
In a rigid container, the volume of the gas remains constant. According to Gay-Lussac's law, the pressure of an ideal gas is directly proportional to its absolute temperature when the volume is constant. This can be expressed as:
\[ P \propto T \]
or
\[ \frac{P_2}{P_1} = \frac{T_2}{T_1} \]

### Step 2: Identify the initial and final temperatures
From the problem, we have:
- Initial temperature \( T_1 = 100 \, K \)
- Final temperature \( T_2 = 109 \, K \)

### Step 3: Set up the equation for pressure
Let the initial pressure be \( P_1 = P \). We need to find the final pressure \( P_2 \). Using the proportionality:
\[ \frac{P_2}{P} = \frac{T_2}{T_1} \]

Substituting the values:
\[ \frac{P_2}{P} = \frac{109 \, K}{100 \, K} \]

### Step 4: Solve for \( P_2 \)
Cross-multiplying gives:
\[ P_2 = P \times \frac{109}{100} = 1.09 P \]

### Step 5: Calculate the change in pressure
The change in pressure \( \Delta P \) can be calculated as:
\[ \Delta P = P_2 - P_1 = 1.09 P - P = 0.09 P \]

### Step 6: Calculate the percentage increase in pressure
The percentage increase in pressure \( X \) is given by:
\[ X = \left( \frac{\Delta P}{P_1} \right) \times 100 \]
Substituting the values:
\[ X = \left( \frac{0.09 P}{P} \right) \times 100 \]

### Step 7: Simplify the expression
The \( P \) cancels out:
\[ X = 0.09 \times 100 = 9 \]

### Final Answer
Thus, the value of \( X \) is:
\[ \boxed{9\%} \]

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