A hydrogen cylinder is designed to withstand an internal pressure of 100 atm . At `27^(@) C` , hydrogen is pumped into the cylinder which exerts a pressure of 20 atm . At which temperature does the danger of explosion first sets in ?

Options

A. 500 K
B. 1500 K
C. 1000 K
D. 2000 K

Answer

To solve the problem, we will use the ideal gas law, which states that for a given amount of gas at constant volume, the pressure is directly proportional to the temperature. We can express this relationship mathematically as:

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

Where:
- \( P_1 \) is the maximum pressure the cylinder can withstand (100 atm),
- \( P_2 \) is the pressure of hydrogen at the initial condition (20 atm),
- \( T_1 \) is the temperature corresponding to \( P_1 \) (unknown),
- \( T_2 \) is the initial temperature (27°C).

### Step 1: Convert the initial temperature from Celsius to Kelvin
To use the ideal gas law, we need to convert the temperature from Celsius to Kelvin. The conversion formula is:

\[ T(K) = T(°C) + 273 \]

So, for \( T_2 \):

\[ T_2 = 27 + 273 = 300 \, K \]

### Step 2: Set up the proportion using the ideal gas law
We know that:

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

Substituting the known values:

\[ \frac{100 \, atm}{20 \, atm} = \frac{T_1}{300 \, K} \]

### Step 3: Simplify the left side of the equation
Calculating the left side:

\[ \frac{100}{20} = 5 \]

So we have:

\[ 5 = \frac{T_1}{300} \]

### Step 4: Solve for \( T_1 \)
Now, we can solve for \( T_1 \):

\[ T_1 = 5 \times 300 \]

Calculating this gives:

\[ T_1 = 1500 \, K \]

### Conclusion
The temperature at which the danger of explosion first sets in is:

\[ T_1 = 1500 \, K \]

### Answer
The answer is \( 1500 \, K \).

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