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 the pressure of a gas is directly proportional to its temperature when the volume and the number of moles of gas are constant.

### Step-by-Step Solution:

1. **Identify Known Values**:
- Maximum pressure the cylinder can withstand (P2) = 100 atm
- Initial pressure of hydrogen (P1) = 20 atm
- Initial temperature (T1) = 27°C

2. **Convert Temperature to Kelvin**:
- To convert Celsius to Kelvin, use the formula:
\[ T(K) = T(°C) + 273 \]
- Therefore,
\[ T1 = 27 + 273 = 300 \text{ K} \]

3. **Use the Ideal Gas Law Relationship**:
- According to the ideal gas law, we can set up the following relationship:
\[ \frac{P1}{T1} = \frac{P2}{T2} \]
- Where P1 is the initial pressure, T1 is the initial temperature, P2 is the maximum pressure, and T2 is the temperature at which the maximum pressure occurs.

4. **Rearranging the Equation to Solve for T2**:
- Rearranging the equation gives us:
\[ T2 = \frac{P2 \times T1}{P1} \]

5. **Substituting the Known Values**:
- Substitute P1, P2, and T1 into the equation:
\[ T2 = \frac{100 \text{ atm} \times 300 \text{ K}}{20 \text{ atm}} \]

6. **Calculating T2**:
- Now calculate T2:
\[ T2 = \frac{30000}{20} = 1500 \text{ K} \]

### Conclusion:
The temperature at which the danger of explosion first sets in is **1500 K**.