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 \( 324^{\circ} \mathrm{K} \).

To solve the problem, we'll use the relationship between pressure \( P \), volume \( V \), and temperature \( T \) given by: \[ P \propto \frac{T}{V} \] This can be expressed as: \[ P = k \frac{T}{V} \] where \( k \) is the proportionality constant. ### Step 1: Establish the Relationship Given two states of the gas: 1. **Initial State:** - Pressure \( P_1 = 10^6 \) Pa - Temperature \( T_1 = 300 \) K - Volume \( V_1 = 1000 \) cm³ 2. **Final State:** - Pressure \( P_2 = 1.8 \times 10^6 \) Pa - Temperature \( T_2 = 324 \) K - Volume \( V_2 = ? \) Using the proportionality: \[ \frac{P_1}{P_2} = \frac{T_1 / V_1}{T_2 / V_2} \] Rearranging to solve for \( V_2 \): \[ V_2 = V_1 \times \frac{P_1 \times T_2}{P_2 \times T_1} \] ### Step 2: Plug in the Known Values \[ V_2 = 1000 \, \text{cm}^3 \times \frac{10^6 \, \text{Pa} \times 324 \, \text{K}}{1.8 \times 10^6 \, \text{Pa} \times 300 \, \text{K}} \] ### Step 3: Simplify the Calculation \[ V_2 = 1000 \times \frac{324}{1.8 \times 300} = 1000 \times \frac{324}{540} = 1000 \times 0.6 = 600 \, \text{cm}^3 \] ### **Final Answer:** When the pressure is \( 1.8 \times 10^6 \) Pa and the temperature is \( 324 \) K, the volume of the gas is **600 cm³**.