A motor car type has a pressure of 2 atmosphere at `27^(@)C`. It suddenly burts. If `C_(P)//C_(V) = 1.4` for air, then the resulting temperature is

Options

A. 27 K
B. `27^(@)C`
C. `-27^(@)C`
D. `246^(@)C`

Explanation

To solve the problem, we need to find the resulting temperature after the tire bursts, using the principles of adiabatic expansion. Here’s a step-by-step solution:

### Step 1: Convert the initial temperature to Kelvin
The initial temperature \(T_1\) is given as \(27^\circ C\). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T_1 = 27 + 273 = 300 \, K \]

**Hint:** Always convert Celsius to Kelvin when dealing with thermodynamic equations.

### Step 2: Identify the initial and final pressures
The initial pressure \(P_1\) is given as \(2 \, atm\) and after the tire bursts, the final pressure \(P_2\) becomes \(1 \, atm\) (since the air expands into the atmosphere).

**Hint:** Remember that when a tire bursts, the pressure inside equalizes with the atmospheric pressure.

### Step 3: Use the adiabatic process equation
For an adiabatic process, we can use the following relationship:
\[ \frac{T_1}{T_2} = \left(\frac{P_1}{P_2}\right)^{\frac{\gamma - 1}{\gamma}} \]
Where \(\gamma = \frac{C_P}{C_V} = 1.4\).

**Hint:** The adiabatic process assumes no heat exchange with the surroundings.

### Step 4: Rearranging the equation to find \(T_2\)
We can rearrange the equation to solve for \(T_2\):
\[ T_2 = T_1 \left(\frac{P_1}{P_2}\right)^{\frac{\gamma - 1}{\gamma}} \]

**Hint:** Isolate the variable you need to find by rearranging the equation.

### Step 5: Substitute the known values into the equation
Substituting the known values:
- \(T_1 = 300 \, K\)
- \(P_1 = 2 \, atm\)
- \(P_2 = 1 \, atm\)
- \(\gamma = 1.4\)

We get:
\[ T_2 = 300 \left(\frac{2}{1}\right)^{\frac{1.4 - 1}{1.4}} = 300 \times 2^{\frac{0.4}{1.4}} \]

**Hint:** Make sure to calculate the exponent carefully.

### Step 6: Calculate the exponent
Calculating the exponent:
\[ \frac{0.4}{1.4} \approx 0.2857 \]
Thus,
\[ T_2 = 300 \times 2^{0.2857} \]

**Hint:** Use a calculator for precise calculations.

### Step 7: Calculate \(2^{0.2857}\)
Using a calculator, we find:
\[ 2^{0.2857} \approx 1.231 \]
So,
\[ T_2 \approx 300 \times 1.231 \approx 369.3 \, K \]

**Hint:** Always round your final answer to a reasonable number of significant figures.

### Step 8: Convert \(T_2\) back to Celsius
To convert \(T_2\) back to Celsius:
\[ T(°C) = T(K) - 273 \]
Thus,
\[ T_2(°C) = 369.3 - 273 \approx 96.3 \, °C \]

**Hint:** Remember to subtract 273 to convert Kelvin back to Celsius.

### Final Answer
The resulting temperature after the tire bursts is approximately \(96.3^\circ C\).