A thin balloon filled with air at `47^(@)C` has a volume of 3 litre. If on placing it in a cooled room its volume becomes 2.7 litre, the temperature of room is:

Options

A. `42^(@)C`
B. `100^(@)C`
C. `15^(@)C`
D. `200^(@)C`

Answer

To solve the problem of finding the temperature of the room in which the balloon is placed, we can use the combined gas law, which relates the volume and temperature of a gas at constant pressure. Here’s a step-by-step solution:

### Step 1: Convert the initial temperature to Kelvin
The initial temperature of the balloon is given as \(47^\circ C\). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T_1 = 47 + 273 = 320 \, K \]

### Step 2: Identify the initial and final volumes
The initial volume \(V_1\) of the balloon is 3 liters, and the final volume \(V_2\) after being placed in the cooled room is 2.7 liters.

### Step 3: Use the combined gas law
According to the combined gas law, we have:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
Where:
- \(V_1 = 3 \, L\)
- \(T_1 = 320 \, K\)
- \(V_2 = 2.7 \, L\)
- \(T_2\) is what we need to find.

### Step 4: Rearrange the equation to solve for \(T_2\)
Rearranging the equation gives us:
\[ T_2 = \frac{V_2 \cdot T_1}{V_1} \]

### Step 5: Substitute the known values
Substituting the known values into the equation:
\[ T_2 = \frac{2.7 \, L \cdot 320 \, K}{3 \, L} \]

### Step 6: Calculate \(T_2\)
Calculating this gives:
\[ T_2 = \frac{864}{3} = 288 \, K \]

### Step 7: Convert \(T_2\) back to Celsius
To convert \(T_2\) back to Celsius, we use the formula:
\[ T(°C) = T(K) - 273 \]
So,
\[ T_2(°C) = 288 - 273 = 15^\circ C \]

### Final Answer
The temperature of the room is \(15^\circ C\).

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