An open flask contains air at 27°C.To what temperature it must be heated to expel one-fourth of the air?

Options

A. 225 K
B. 65°C
C. 927°C
D. 460°C

Answer

To solve the problem of determining the temperature to which the air in an open flask must be heated to expel one-fourth of the air, we can follow these steps:

### Step 1: Understand the Initial Conditions
The initial temperature of the air in the flask is given as 27°C. We need to convert this temperature to Kelvin for our calculations.

**Hint:** Remember that to convert Celsius to Kelvin, you add 273.15 to the Celsius temperature.

**Calculation:**
\[ T_1 = 27°C + 273.15 = 300.15 \, K \]

### Step 2: Determine the Amount of Air Remaining
If one-fourth of the air is expelled, then three-fourths of the air remains in the flask. If we denote the initial amount of air as \( n \), then the remaining amount of air after expelling one-fourth is:

**Hint:** Think about how much air is left after removing a fraction of it.

**Calculation:**
\[ n_2 = n - \frac{n}{4} = \frac{3n}{4} \]

### Step 3: Apply the Ideal Gas Law Relationship
Since the volume and pressure are constant, we can use the relationship between the number of moles and temperature. According to the ideal gas law, we can express this as:

\[ n_1 T_1 = n_2 T_2 \]

Where:
- \( n_1 \) is the initial number of moles (which is \( n \)),
- \( T_1 \) is the initial temperature (300 K),
- \( n_2 \) is the remaining number of moles (which is \( \frac{3n}{4} \)),
- \( T_2 \) is the final temperature we need to find.

**Hint:** This equation relates the initial and final states of the gas under constant volume and pressure.

### Step 4: Substitute Values into the Equation
Now, substitute the known values into the equation:

**Calculation:**
\[ n \cdot 300 = \frac{3n}{4} \cdot T_2 \]

### Step 5: Solve for \( T_2 \)
We can simplify the equation by canceling \( n \) from both sides (assuming \( n \neq 0 \)):

**Hint:** Isolate \( T_2 \) on one side of the equation.

**Calculation:**
\[ 300 = \frac{3}{4} T_2 \]
\[ T_2 = \frac{300 \cdot 4}{3} \]
\[ T_2 = 400 \, K \]

### Step 6: Convert \( T_2 \) Back to Celsius
Finally, we can convert the temperature back to Celsius if needed:

**Hint:** To convert from Kelvin back to Celsius, subtract 273.15.

**Calculation:**
\[ T_2 = 400 - 273.15 = 126.85°C \]

### Final Answer
To expel one-fourth of the air from the flask, the temperature must be heated to approximately **400 K** or **126.85°C**.

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