To solve the problem of finding the ratio of the final volume to the initial volume of a gas when its pressure and temperature change, we can use the Ideal Gas Law, which is expressed as: \[ PV = nRT \] Where: - \( P \) = Pressure - \( V \) = Volume - \( n \) = Number of moles of gas (constant in this case) - \( R \) = Ideal gas constant (constant) - \( T \) = Temperature in Kelvin ### Step 1: Convert Temperatures to Kelvin First, we need to convert the temperatures from Celsius to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \] - Initial temperature: \[ T_1 = 127°C + 273.15 = 400.15 K \] - Final temperature: \[ T_2 = -73°C + 273.15 = 200.15 K \] ### Step 2: Apply the Ideal Gas Law Since the number of moles of gas and the gas constant remain constant, we can set up the relationship between the initial and final states of the gas: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \] Rearranging this gives us: \[ \frac{V_2}{V_1} = \frac{P_1 T_2}{P_2 T_1} \] ### Step 3: Substitute the Known Values Now we can substitute the known values into the equation: - \( P_1 = 300 \, \text{Nm}^{-2} \) - \( P_2 = 120 \, \text{Nm}^{-2} \) - \( T_1 = 400.15 \, K \) - \( T_2 = 200.15 \, K \) Substituting these values into the equation gives: \[ \frac{V_2}{V_1} = \frac{300 \times 200.15}{120 \times 400.15} \] ### Step 4: Calculate the Ratio Now we perform the calculations step-by-step: 1. Calculate the numerator: \[ 300 \times 200.15 = 60045 \] 2. Calculate the denominator: \[ 120 \times 400.15 = 48018 \] 3. Now, calculate the ratio: \[ \frac{V_2}{V_1} = \frac{60045}{48018} \approx 1.25 \] ### Step 5: Express the Ratio in Simplified Form To express \( 1.25 \) as a ratio, we can write it as: \[ \frac{V_2}{V_1} = \frac{5}{4} \] This means that the final volume \( V_2 \) is \( \frac{5}{4} \) times the initial volume \( V_1 \). ### Conclusion: Correct Option Thus, the ratio of the final volume to the initial volume is: **B. 5 : 4** ### Explanation of Other Options - **A. 2 : 5**: This suggests that the final volume is less than the initial volume, which contradicts our calculation showing an increase in volume. - **C. 4 : 5**: This implies that the final volume is less than the initial volume, which is incorrect based on our calculations. - **D. 5 : 2**: This suggests a much larger increase in volume than what we calculated, which is not supported by the data. ### Revision Summary - Use the Ideal Gas Law to relate pressure, volume, and temperature. - Convert temperatures from Celsius to Kelvin for calculations. - Rearrange the Ideal Gas Law to find the volume ratio. - Carefully perform calculations to avoid errors in the final ratio.