# Calculating Work in Isothermal Gas Expansion with Variable R_v

• alexialight
In summary, Daniel found that the work done by the gas during the isothermal expansion (path C) is equal to the area under the curve of the graph of p vs. V.
alexialight
Calculate the work W done by the gas during the isothermal expansion (path C). It may be be convenient to generalize your results by using the variable R_v, which is the ratio of final to initial volumes (equal to 4 for the expansions shown in the figure.)
Express W in terms of p_0, V_0, and R_v.

I've been at this question for ages and I just can't see how p_0 fits into the answer. The hints say to find an expression for p(V) in terms of p_0, V_0 and V and I can't even seem to do that. Any help would be much appreciated.

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U can find it writing Mendeleev-Clapeyron's equation for the initial & final states.

U have to integrate it to get the answer.

Daniel.

Knowing that

$$W = \int_{V_1}^{V_2} p ~dV$$

and

$$p = \frac {nRT}{V}$$

you can get

$$W=nRT \ln \frac {V_2}{V_1}$$

Since it is an isoterm, then $T$ is constant.

$$p_1V_1 = p_2 V_2$$ or

$$\frac {V_2}{V_1} = R_V = \frac {p_1}{p_2}$$

This is a MP problem isn't it?

Yes it is :(
I understand that W is equivilent to that equation you wrote, I just don't know how to get an answer with only p_0, V_0 and R_v and not n, T and R

Well If I recall correctly, just because MP says you have to use these variables, it doesn't mean all the variables have to be used. But you can't use variables that aren't defined for the problem.

So you agree that

$$W=nRT \ln R_v$$

Since n, R, T are all constants in this problem.

$$p_0V_0 = nRT = pV$$, then perhaps your final answer is

$$W=nRT \ln R_v = p_0 V_0 \ln R_v$$

Enter that on your own risk, I never liked MP. Good luck.

I had the same problem. It works now though :)

## What is a PV plot and why is it important in gas expansion studies?

A PV plot, or pressure-volume plot, is a graph that shows the relationship between the pressure and volume of a gas. It is important in gas expansion studies because it allows scientists to visualize how gases behave under different temperatures and pressures, and can help them understand the properties of gases.

## How is a PV plot constructed?

A PV plot is constructed by measuring the pressure and volume of a gas at different points and plotting them on a graph. The pressure is typically measured using a manometer or pressure gauge, and the volume can be measured using a syringe or other device. The data points are then connected with a line to create the PV plot.

## What does a PV plot tell us about gas expansion?

A PV plot can tell us about the behavior of a gas during expansion. For example, if the pressure remains constant while the volume increases, this indicates that the gas is undergoing isobaric expansion. If the temperature remains constant during expansion, this indicates isothermal expansion. The slope of the line on the PV plot can also tell us about the compressibility of the gas.

## How does temperature affect a PV plot?

The temperature of a gas can affect its PV plot in several ways. An increase in temperature can cause the graph to shift upwards, indicating an increase in pressure. It can also cause the slope of the line to decrease, indicating that the gas is becoming less compressible. Additionally, changes in temperature can cause the shape of the graph to change, indicating a change in the type of expansion (isobaric, isothermal, etc.)

## What are some real-world applications of PV plots in gas expansion studies?

PV plots have many practical applications in fields such as chemistry, physics, and engineering. They can be used to study the behavior of gases in various industrial processes, such as in the design of engines or refrigeration systems. They are also used in weather forecasting and in the study of atmospheric gases. Additionally, PV plots are important in understanding the behavior of gases in biological systems, such as in the lungs during respiration.

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