The pressure inside a tyre is 4 times that of atmosphere. If the tyre bursts suddenly at temperature. 300 K. What will be the new temperature ?

Options

A. `300 (4)^(7//2)`
B. `300 (4)^(2//7)`
C. `300 (2)^(7//2)`
D. `300 (4)^(-2//7)`

Explanation

To solve the problem, we need to apply the principles of thermodynamics, specifically the relationship between pressure and temperature during an adiabatic process. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- The pressure inside the tyre (P1) is 4 times the atmospheric pressure (P0).
- Therefore, P1 = 4 * P0.
- The initial temperature (T1) is given as 300 K.

### Step 2: Identify the Final Pressure (P2)
- When the tyre bursts, the pressure inside the tyre becomes equal to the atmospheric pressure.
- Thus, P2 = P0.

### Step 3: Use the Adiabatic Process Formula
For an adiabatic process, the relationship between temperatures and pressures is given by:
\[ \frac{T_2}{T_1} = \left(\frac{P_1}{P_2}\right)^{\frac{1 - \gamma}{\gamma}} \]
where:
- \( T_1 \) = initial temperature
- \( T_2 \) = final temperature
- \( P_1 \) = initial pressure
- \( P_2 \) = final pressure
- \( \gamma \) = heat capacity ratio (for air, \( \gamma = \frac{7}{5} \))

### Step 4: Substitute the Values
Substituting the known values into the formula:
\[ \frac{T_2}{300} = \left(\frac{4P_0}{P_0}\right)^{\frac{1 - \frac{7}{5}}{\frac{7}{5}}} \]
This simplifies to:
\[ \frac{T_2}{300} = 4^{\frac{1 - \frac{7}{5}}{\frac{7}{5}}} \]
Calculating \( 1 - \frac{7}{5} = -\frac{2}{5} \), we rewrite the equation:
\[ \frac{T_2}{300} = 4^{-\frac{2}{7}} \]

### Step 5: Calculate \( T_2 \)
Now, we can express \( T_2 \):
\[ T_2 = 300 \times 4^{-\frac{2}{7}} \]
Calculating \( 4^{-\frac{2}{7}} \):
- \( 4^{-\frac{2}{7}} = \left(2^2\right)^{-\frac{2}{7}} = 2^{-\frac{4}{7}} \)

Now, substituting this back:
\[ T_2 = 300 \times 2^{-\frac{4}{7}} \]

### Step 6: Final Calculation
To find the numerical value of \( T_2 \):
1. Calculate \( 2^{-\frac{4}{7}} \) using a calculator or logarithmic tables.
2. Multiply the result by 300 to get the final temperature \( T_2 \).

### Conclusion
After performing the calculations, you will find the new temperature \( T_2 \).