What will be the final pressure of an ideal gas present in a cylinder at 2 atm when the temperature of the gas is increased from `100^(@)C` to `500^(@)C`?

Options

A. 3.68 atm
B. 4.14 atm
C. 1.77 atm
D. 8.42 atm

Answer

To find the final pressure of an ideal gas when the temperature is increased, we can use the relationship derived from the ideal gas law, which states that for a constant volume and number of moles, the ratio of pressure to temperature remains constant. This can be expressed as:

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

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Initial Pressure, \( P_1 = 2 \, \text{atm} \)
- Initial Temperature, \( T_1 = 100^\circ C \)

2. **Convert Initial Temperature to Kelvin:**
- To convert Celsius to Kelvin, use the formula:
\[ T(K) = T(°C) + 273 \]
- Therefore,
\[ T_1 = 100 + 273 = 373 \, K \]

3. **Identify the Final Temperature:**
- Final Temperature, \( T_2 = 500^\circ C \)

4. **Convert Final Temperature to Kelvin:**
- Using the same conversion formula:
\[ T_2 = 500 + 273 = 773 \, K \]

5. **Set Up the Equation:**
- Using the relationship:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
- Substitute the known values:
\[ \frac{2 \, \text{atm}}{373 \, K} = \frac{P_2}{773 \, K} \]

6. **Solve for \( P_2 \):**
- Rearranging the equation to find \( P_2 \):
\[ P_2 = \frac{2 \, \text{atm} \times 773 \, K}{373 \, K} \]
- Calculate \( P_2 \):
\[ P_2 = \frac{1546 \, \text{atm} \cdot K}{373 \, K} \approx 4.14 \, \text{atm} \]

7. **Final Answer:**
- The final pressure \( P_2 \) is approximately \( 4.14 \, \text{atm} \).