By what amount must the temperature of 200cm$^3$ of Nitrogen at 27°C be increased to double the pressure if the final volume is 150cm$^3$ (Assume ideality)

Analysis

To solve this problem, we use the General Gas Law (Combined Gas Law), which relates pressure ($P$), volume ($V$), and temperature ($T$) for a fixed mass of an ideal gas:

Concept

• Initial state: $V_1 = 200 \text{ cm}^3$, $T_1 = 27^\circ\text{C}$, $P_1 = P$
• Final state: $V_2 = 150 \text{ cm}^3$, $P_2 = 2P$ (pressure is doubled), $T_2 = ?$
• Crucial Rule: Always convert temperature from Celsius to Kelvin by adding 273. • $T_1 = 27 + 273 = 300 \text{ K}$

Step-by-Step Solution

1. Set up the equation: $$\frac{P \times 200}{300} = \frac{2P \times 150}{T_2}$$

2. Simplify the equation: The pressure $P$ appears on both sides, so it cancels out: $$\frac{200}{300} = \frac{300}{T_2}$$ $$\frac{2}{3} = \frac{300}{T_2}$$

3. Solve for $T_2$: $$2 \times T_2 = 3 \times 300$$ $$2T_2 = 900$$ $$T_2 = 450 \text{ K}$$

4. Convert back to Celsius: $$T_2 (\text{in } ^\circ\text{C}) = 450 - 273 = 177^\circ\text{C}$$

5. Calculate the increase in temperature: The question asks for the amount by which the temperature must be increased: $$\text{Increase} = T_2 - T_1$$ $$\text{Increase} = 177^\circ\text{C} - 27^\circ\text{C} = 150^\circ\text{C}$$

Conclusion

The temperature must be increased by 150°C to reach the final temperature of 177°C.

Exam Tip: Always check if the question asks for the final temperature or the increase in temperature. Many students stop at 177°C and choose option B, but the question specifically asks for the "amount... increased," which is $177 - 27 = 150$.