To what temperature must a gas at 273k be heated in order to double both its volume and pressure?

Analysis

This question tests your understanding of the General Gas Law, which combines Boyle’s Law, Charles’s Law, and the Pressure Law. The formula is:

Concept

The General Gas Law states that for a fixed mass of gas, the ratio of the product of pressure and volume to the temperature remains constant.

• $P_1, V_1, T_1$ are the initial pressure, volume, and temperature.
• $P_2, V_2, T_2$ are the final pressure, volume, and temperature.

Step-by-Step Solution

1. Identify the given values: • Initial temperature ($T_1$) = $273\text{ K}$ • Initial pressure = $P_1$ • Initial volume = $V_1$

2. Identify the final conditions: • The pressure is doubled: $P_2 = 2P_1$ • The volume is doubled: $V_2 = 2V_1$ • Final temperature ($T_2$) = ?

3. Substitute these into the formula: $$\frac{P_1 \times V_1}{273} = \frac{(2P_1) \times (2V_1)}{T_2}$$

4. Simplify the equation: $$\frac{P_1V_1}{273} = \frac{4P_1V_1}{T_2}$$

5. Cancel out $P_1$ and $V_1$ from both sides: $$\frac{1}{273} = \frac{4}{T_2}$$

6. Solve for $T_2$: $$T_2 = 273 \times 4$$ $$T_2 = 1092\text{ K}$$

Conclusion

By doubling both the pressure and the volume, the temperature must increase by a factor of $2 \times 2 = 4$. Multiplying the initial temperature of $273\text{ K}$ by $4$ gives $1092\text{ K}$.

Exam Tip: When dealing with gas law problems, always ensure your temperature is in Kelvin. If a question gives you temperature in Celsius, add $273$ to it before plugging it into your formula. Also, remember that if both $P$ and $V$ increase, the temperature must increase significantly to maintain the balance!