Boyle's Law states that for a fixed mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. Mathematically, this is expressed as P1 V1 = P2 V2.
This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Here is the solution for question 2(b), which applies Boyle's Law.
Boyle's Law states that for a fixed mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. Mathematically, this is expressed as $P_1 V_1 = P_2 V_2$.
Initial conditions:
Volume, $V_1 = 40 \text{ cm}^3$
Temperature, $T_1 = 0^\circ C$
Pressure, $P_1 = 680 \text{ mmHg}$
Final conditions for part (b):
Temperature, $T_2 = 0^\circ C$ (constant)
Pressure, $P_2 = 380 \text{ mmHg}$
Volume, $V_2 = ?$
Step 1: Identify the given values.
$$P_1 = 680 \text{ mmHg}$$
$$V_1 = 40 \text{ cm}^3$$
$$P_2 = 380 \text{ mmHg}$$
Since the temperature is constant ($T_1 = T_2 = 0^\circ C$), we can use Boyle's Law.
Step 2: Apply Boyle's Law formula.
$$P_1 V_1 = P_2 V_2$$
Step 3: Rearrange the formula to solve for $V_2$.
$$V_2 = \frac{P_1 V_1}{P_2}$$
Step 4: Substitute the values and calculate.
$$V_2 = \frac{(680 \text{ mmHg}) \times (40 \text{ cm}^3)}{380 \text{ mmHg}}$$
$$V_2 = \frac{27200 \text{ cm}^3}{380}$$
$$V_2 = 71.5789... \text{ cm}^3$$
Step 5: Round the answer to an appropriate number of significant figures.
$$V_2 \approx 71.6 \text{ cm}^3$$
The volume the gas would occupy at $0^\circ C$ and $380 \text{ mmHg}$ pressure is $\boxed{\text{71.6 cm}^3}$.
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