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

Answer

To solve the problem, we need to determine the temperatures at which one-third of the air escapes from an open flask at 27°C. We will break this down into two parts as per the question.

### Part (a): Calculate the temperature at which 1/3 of air measured at 27°C escapes out.

1. **Convert the initial temperature to Kelvin**:
\[ T_1 = 27°C = 27 + 273 = 300 \, K \]

2. **Determine the initial number of moles of air**:
Let the initial number of moles of air be \( n \).

3. **Calculate the number of moles remaining after 1/3 escapes**:
If 1/3 of the air escapes, then the remaining air is:
\[ n - \frac{n}{3} = \frac{2n}{3} \]

4. **Using the ideal gas law at constant pressure**:
According to the ideal gas law, we can relate the number of moles and temperature:
\[ n_1 T_1 = n_2 T_2 \]
Where:
- \( n_1 = n \) (initial moles)
- \( T_1 = 300 \, K \)
- \( n_2 = \frac{2n}{3} \) (remaining moles)
- \( T_2 \) is the final temperature we need to find.

5. **Substituting values into the equation**:
\[ n \cdot 300 = \frac{2n}{3} \cdot T_2 \]

6. **Cancel \( n \) from both sides** (assuming \( n \neq 0 \)):
\[ 300 = \frac{2}{3} T_2 \]

7. **Solve for \( T_2 \)**:
\[ T_2 = 300 \cdot \frac{3}{2} = 450 \, K \]

8. **Convert \( T_2 \) back to Celsius**:
\[ T_2 = 450 - 273 = 177°C \]

### Part (b): Calculate the temperature at which 1/3 of air measured at final temperature escapes out.

1. **Initial moles before heating**:
Let the initial number of moles of air be \( n \).

2. **After heating, if 1/3 escapes**:
The remaining moles would be:
\[ n - \frac{n}{3} = \frac{2n}{3} \]

3. **Using the ideal gas law again**:
Now, we will use the final temperature \( T_2 \) from part (a) which is 450 K:
\[ n_1 T_1 = n_2 T_2 \]
Where:
- \( n_1 = \frac{n}{3} \) (initial moles before heating)
- \( T_1 \) is what we need to find.
- \( n_2 = \frac{2n}{3} \) (remaining moles)
- \( T_2 = 450 \, K \)

4. **Substituting values into the equation**:
\[ \frac{n}{3} \cdot T_1 = \frac{2n}{3} \cdot 450 \]

5. **Cancel \( n \) from both sides**:
\[ \frac{1}{3} T_1 = \frac{2}{3} \cdot 450 \]

6. **Solve for \( T_1 \)**:
\[ T_1 = 2 \cdot 450 = 900 \, K \]

7. **Convert \( T_1 \) back to Celsius**:
\[ T_1 = 900 - 273 = 627°C \]

### Final Answers:
- (a) The temperature at which 1/3 of air escapes is **177°C**.
- (b) The temperature at which 1/3 of air measured at final temperature escapes is **627°C**.