An iron cylinder contains helium at a pressure of 250 k Pa at 300 K. The cylinder can withstand a pressure of `1 xx 10^6` Pa. The room in which cylinder is placed catches fire. Predict the temperature (in K) at which the cylinder will blow up before it melts or not (m.p.t. of the cylinder =1800K)..

Explanation

To solve the problem, we will use the ideal gas law, which states that for a given mass of gas at constant volume, the pressure is directly proportional to the temperature. This can be expressed as:

\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]

### Step 1: Identify the given values
- Initial pressure, \( P_1 = 250 \, \text{kPa} = 250 \times 10^3 \, \text{Pa} \)
- Initial temperature, \( T_1 = 300 \, \text{K} \)
- Final pressure, \( P_2 = 1 \times 10^6 \, \text{Pa} \)

### Step 2: Set up the equation using the ideal gas law
Using the relationship from the ideal gas law, we can rearrange it to solve for \( T_2 \):

\[ T_2 = \frac{P_2 \cdot T_1}{P_1} \]

### Step 3: Substitute the known values into the equation
Now we can substitute the known values into the equation:

\[ T_2 = \frac{(1 \times 10^6 \, \text{Pa}) \cdot (300 \, \text{K})}{(250 \times 10^3 \, \text{Pa})} \]

### Step 4: Calculate \( T_2 \)
Now, perform the calculation:

\[ T_2 = \frac{(1 \times 10^6) \cdot 300}{250 \times 10^3} \]

Calculating the numerator:

\[ 1 \times 10^6 \cdot 300 = 300 \times 10^6 \]

Calculating the denominator:

\[ 250 \times 10^3 = 250000 \]

Now divide:

\[ T_2 = \frac{300 \times 10^6}{250000} = \frac{300000000}{250000} = 1200 \, \text{K} \]

### Step 5: Conclusion
The calculated temperature \( T_2 \) at which the cylinder will blow up is \( 1200 \, \text{K} \). Since this temperature is below the melting point of the iron cylinder (1800 K), the cylinder will blow up before it melts.

### Summary
- The cylinder will blow up at a temperature of \( 1200 \, \text{K} \) before it melts.

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