An open flask contains air at `27^(@)C` Calculate the temperature at which it should be heated so that `(1)/(3)` rd of air measured at `27^(@)C` escapes out

Options

A. (a) `177^(@)C`
B. (b) `100^(@)C`
C. (c) `300^(@)C`
D. (d) `150^(@)C`

Explanation

To solve the problem step by step, we will use the Ideal Gas Law and the concept of moles of gas.

### Step 1: Understand the problem
We have an open flask containing air at an initial temperature of \(27^\circ C\). We need to find the temperature at which one-third of the air escapes.

### Step 2: Determine the initial conditions
- Initial temperature (\(T_1\)) = \(27^\circ C\)
- Convert \(T_1\) to Kelvin:
\[ T_1 = 27 + 273 = 300 \, K \]

### Step 3: Calculate the initial number of moles
Let the initial number of moles of air be \(n\).

### Step 4: Determine the final number of moles
If one-third of the air escapes, the remaining moles will be:
\[ n_2 = n - \frac{1}{3}n = \frac{2}{3}n \]

### Step 5: Apply the Ideal Gas Law
According to the Ideal Gas Law, we can express the relationship between the initial and final states of the gas:
\[ n_1 T_1 = n_2 T_2 \]
Where:
- \(n_1 = n\)
- \(n_2 = \frac{2}{3}n\)
- \(T_1 = 300 \, K\)
- \(T_2\) is the final temperature we need to find.

### Step 6: Substitute the values into the equation
Substituting the known values into the equation:
\[ n \cdot 300 = \left(\frac{2}{3}n\right) T_2 \]
We can cancel \(n\) from both sides (assuming \(n \neq 0\)):
\[ 300 = \frac{2}{3} T_2 \]

### Step 7: Solve for \(T_2\)
To find \(T_2\), we rearrange the equation:
\[ T_2 = \frac{300 \cdot 3}{2} = 450 \, K \]

### Step 8: Convert \(T_2\) back to Celsius
To convert \(T_2\) from Kelvin to Celsius:
\[ T_2 = 450 - 273 = 177^\circ C \]

### Final Answer
The temperature at which the air should be heated so that one-third escapes is \(177^\circ C\).

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