5. The pressure of a gas is inversely proportional to its volume and directly proportional to its absolute temperature. When the pressure is \( 10^{6} \) pa and the temperature \( 300^{\circ} \mathrm{K} \), the volume is \( 1000 \mathrm{~cm}^{3} \). Find the volume when the pressure is \( 1.8 \times 10^{6} \) pa and the temperature \( 324^{\circ} \mathrm{K} \).
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To find the new volume, we can use the Ideal Gas Law which states that \( P \times V = n \times R \times T \). Given that the pressure is inversely proportional to the volume and directly proportional to the temperature, we can set up a relationship based on the initial and final states. Using the initial conditions: - \( P_1 = 10^{6} \) pa - \( V_1 = 1000 \) cm³ (or \( 1 \) L) - \( T_1 = 300 \) K And the new conditions: - \( P_2 = 1.8 \times 10^{6} \) pa - \( T_2 = 324 \) K According to the proportions: \[ \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2} \] Plugging in the values we have: \[ \frac{10^{6} \cdot 1000}{300} = \frac{1.8 \times 10^{6} \cdot V_2}{324} \] Cross-multiplying gives: \[ 10^{6} \cdot 1000 \cdot 324 = 1.8 \times 10^{6} \cdot V_2 \cdot 300 \] Simplifying this: \[ 324 \times 10^{9} = 540 \times 10^{6} \cdot V_2 \] From that, we can find \( V_2 \): \[ V_2 = \frac{324 \times 10^{9}}{540 \times 10^{6}} \approx 600 \text{ cm}^3 \] So the volume when the pressure is \( 1.8 \times 10^{6} \) pa and the temperature is \( 324 \) K is approximately \( 600 \text{ cm}^3 \).
