An ideal gas has an initial pressure of 3 pressure units and an initial volume of 4 volume units. the table gives the final pressure and volume of the gas (in those same units) in five processes. Which processes start and end on the same isotherm?

^{a b c d e}

^{__|__________________________}

^{p | 12 6 5 4 1}

^{V | 1 2 7 3 12}

To determine which processes start and end on the same isotherm for the ideal gas, we need to understand the relationship between pressure, volume, and temperature as described by the ideal gas law. The ideal gas law states that PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature. For a given amount of gas at a constant temperature, the product of pressure and volume (PV) remains constant. This means that if a process occurs at constant temperature, the initial and final states will lie on the same isotherm, which can be visualized on a PV diagram.

Analyzing the Given Data

We have the following initial conditions:

• Initial Pressure (P_initial) = 3 pressure units
• Initial Volume (V_initial) = 4 volume units

The initial state can be represented as:

P_initial × V_initial = 3 × 4 = 12 pressure-volume units.

Final States from the Table

Now, let's look at the final states for each process:

• Process A: P = 12, V = 1 → PV = 12 × 1 = 12
• Process B: P = 6, V = 2 → PV = 6 × 2 = 12
• Process C: P = 5, V = 7 → PV = 5 × 7 = 35
• Process D: P = 4, V = 3 → PV = 4 × 3 = 12
• Process E: P = 1, V = 12 → PV = 1 × 12 = 12

Identifying Processes on the Same Isotherm

From our calculations, we can see that processes A, B, D, and E all have a final state where the product of pressure and volume equals 12. This means that these processes start and end on the same isotherm as the initial state. In contrast, process C has a PV product of 35, which indicates it does not lie on the same isotherm.

Summary of Processes

To summarize, the processes that start and end on the same isotherm are:

• Process A
• Process B
• Process D
• Process E

In conclusion, understanding the relationship between pressure, volume, and temperature is crucial for analyzing gas behavior in different processes. By applying the ideal gas law, we can easily determine which processes maintain the same thermal state, or isotherm, throughout their transitions.