When the temperature of a gas in a metal is incrased from `27^(@)C` to `87^(@)C` the initial pressure of 2 atmospheres changes to

Options

A. 1.0 atm
B. 1.6 atm
C. 2.4 atm
D. 3 atm

Explanation

To solve the problem, we will use the ideal gas law, which states that the pressure of a gas is directly proportional to its temperature when the volume is constant. We can express this relationship mathematically as:

\[ \frac{P_2}{P_1} = \frac{T_2}{T_1} \]

### Step-by-Step Solution:

**Step 1: Convert the temperatures from Celsius to Kelvin.**

- The initial temperature \( T_1 \) is \( 27^\circ C \).
- To convert to Kelvin, use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T_1 = 27 + 273 = 300 \, K \]

- The final temperature \( T_2 \) is \( 87^\circ C \).
- Similarly, convert to Kelvin:
\[ T_2 = 87 + 273 = 360 \, K \]

**Step 2: Identify the initial pressure.**

- The initial pressure \( P_1 \) is given as \( 2 \, atm \).

**Step 3: Use the ideal gas law relationship to find the final pressure \( P_2 \).**

- From the relationship \( \frac{P_2}{P_1} = \frac{T_2}{T_1} \), we can rearrange to find \( P_2 \):
\[ P_2 = P_1 \times \frac{T_2}{T_1} \]

**Step 4: Substitute the known values into the equation.**

- Substitute \( P_1 = 2 \, atm \), \( T_2 = 360 \, K \), and \( T_1 = 300 \, K \):
\[ P_2 = 2 \, atm \times \frac{360 \, K}{300 \, K} \]

**Step 5: Simplify the equation.**

- Calculate the fraction:
\[ P_2 = 2 \, atm \times \frac{360}{300} = 2 \, atm \times 1.2 \]

**Step 6: Calculate the final pressure.**

- Now, calculate \( P_2 \):
\[ P_2 = 2.4 \, atm \]

### Final Answer:
The final pressure \( P_2 \) when the temperature is increased to \( 87^\circ C \) is **2.4 atm**.

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