Problem 61 - Chemistry: An Atoms-Focused Approach

Question

What happens to the volume of gas in a cylinder with a movable piston under the following conditions? a. Both the absolute temperature and the external pressure on the piston double. b. The absolute temperature is halved, and the external pressure on the piston doubles. c. The absolute temperature increases by \(75 \%,\) and the external pressure on the piston increases by \(50 \%\).

Step-by-Step Solution

Answer

a) Doubling both the absolute temperature and the external pressure on the piston. b) Halving the absolute temperature and doubling the external pressure on the piston. c) Increasing absolute temperature by 75% and increasing the external pressure on the piston by 50%. Answer: a) The volume remains the same. b) The volume becomes 25% of the original volume. c) The new volume is 7/6 of the original volume.

Step 1: Condition a: Doubling both the absolute temperature and the external pressure on the piston

We start with the ideal gas law: \(PV = nRT\) For the initial state (P1, V1, T1), we have \(P_1V_1 = nR T_1\). When both the temperature and pressure are doubled, the final state (P2, V2, T2) will give us \(P_2V_2 = nR T_2\). Since P2 = 2P1 and T2 = 2T1, we can express this as \((2P_1)V_2 = nR(2T_1)\). Dividing the initial state equation and the final state equation, we get: \(\frac{(2P_1)V_2}{P_1V_1} = \frac{nR(2T_1)}{nR T_1}\) Simplifying the equation, we get: \(\frac{V_2}{V_1} = 1\). This means that the volume remains the same in this case.

Step 2: Condition b: Halving the absolute temperature and doubling the external pressure on the piston

In this case, T2 = 0.5T1 and P2 = 2P1. Using the initial and final state equations, \(P_1V_1 = nR T_1\) and \((2P_1)V_2 = nR(0.5T_1)\). Dividing the initial state equation and the final state equation, we get: \(\frac{(2P_1)V_2}{P_1V_1} = \frac{nR(0.5T_1)}{nR T_1}\) Simplifying the equation, we get: \(\frac{V_2}{V_1} = 0.25\) This means that the volume is now 25% of the original volume under these conditions.

Step 3: Condition c: Increasing absolute temperature by 75% and increasing the external pressure on the piston by 50%

In this case, T2 = 1.75T1 and P2 = 1.5P1. Using the initial and final state equations, \(P_1V_1 = nR T_1\) and \((1.5P_1)V_2 = nR(1.75T_1)\). Dividing the initial state equation and the final state equation, we get: \(\frac{(1.5P_1)V_2}{P_1V_1} = \frac{nR(1.75T_1)}{nR T_1}\) Simplifying the equation, we get: \(\frac{V_2}{V_1} = \frac{1.75}{1.5} = \frac{7}{6}\) This means that the volume has increased by \(\frac{1}{6}\), so the new volume is \(\frac{7}{6}\) of the original volume under these conditions.