At a constant pressure, what should be the percentage increase in the temperature in kelvin for a 10 % increase in the volume

Options

A. 0.1
B. 0.2
C. 0.05
D. 0.5

Explanation

To solve the problem, we will use the ideal gas law, which states that:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles of gas
- \( R \) = Ideal gas constant
- \( T \) = Temperature in Kelvin

Given that the pressure \( P \) is constant, we can analyze the relationship between volume \( V \) and temperature \( T \).

### Step 1: Understand the relationship between volume and temperature
From the ideal gas law, if \( P \) is constant, we can express the relationship between volume and temperature as:

\[ V \propto T \]

This means that volume is directly proportional to temperature when pressure is constant.

### Step 2: Calculate the increase in volume
We are given that the volume increases by 10%. If the initial volume is \( V \), the new volume \( V' \) can be expressed as:

\[ V' = V + 0.10V = 1.10V \]

### Step 3: Relate the increase in volume to the increase in temperature
Since volume is directly proportional to temperature, we can express the new temperature \( T' \) in terms of the initial temperature \( T \):

\[ T' \propto V' \]

Thus, we can write:

\[ T' = k \cdot V' \]

Where \( k \) is a proportionality constant. Since \( V' = 1.10V \), we can substitute:

\[ T' = k \cdot (1.10V) \]

### Step 4: Express the new temperature in terms of the initial temperature
Since \( T = k \cdot V \), we can express \( T' \) as:

\[ T' = 1.10 \cdot T \]

### Step 5: Calculate the percentage increase in temperature
To find the percentage increase in temperature, we can use the formula:

\[ \text{Percentage Increase} = \left( \frac{T' - T}{T} \right) \times 100\% \]

Substituting \( T' = 1.10T \):

\[ \text{Percentage Increase} = \left( \frac{1.10T - T}{T} \right) \times 100\% \]

\[ = \left( \frac{0.10T}{T} \right) \times 100\% \]

\[ = 10\% \]

### Final Answer
The percentage increase in temperature for a 10% increase in volume at constant pressure is **10%**.

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