Isothermal expansion of an ideal gas

[thezac11]

Homework Statement

One mole of monatomic ideal gas is taken from an initial pressure(P) and volume(V) to a final pressure(2P) and volume(2V). It goes from pressure=P and volume=V to pressure=P/2 and volume=2V through isothermal expansion and from there volume stays constant but the pressure goes to 2P from the previous P/2 and this path is through isochoric compression. The temperature remains constant.

-What is the change in energy of the process?
-What is the heat input?
-What is the work output?
-What is the change in entropy?

Homework Equations

change in energy = (heat in) - (work out)

change in entropy = heat in/temperature

Ideal gas law: PV=nRT, where n=amount of substance, R=constant, T=temp.

The Attempt at a Solution

I know the answers I just want to know how to get them. If you can explain the process I would greatly appreciate it. The final answers are:

-Change in energy = 0

-Heat input = P x V x (ln2)

-Work out = P x V x (ln2)

-Change in entropy = R(ln2)

 

[Andrew Mason]

Homework Statement

One mole of monatomic ideal gas is taken from an initial pressure(P) and volume(V) to a final pressure(2P) and volume(2V). It goes from pressure=P and volume=V to pressure=P/2 and volume=2V through isothermal expansion and from there volume stays constant but the pressure goes to 2P from the previous P/2 and this path is through isochoric compression. The temperature remains constant.

-What is the change in energy of the process?

I am not sure what this question is asking. There are three forms of energy: Work, Heat flow and internal energy.

The change in internal energy

What is the equation for $\Delta U = ?$.(Hint: it involves Cv). What is the Cv for this gas?

Determine the change in U in each step. There is no change in U in the first step (isothermal expansion) but there is a change in internal energy for the second step (isochoric heating). You will have to work out what that is.

-What is the heat input?

In the first step, there is no change in U. So what is the relationship between $\Delta Q$ and W for the first step? In the second step, what is the relationship between P and T if V is constant? If P increases by 4x what happens to T? What amount of heat is required to raise the temperature by that amount (hint: it involves Cv).

-What is the work output?

What is the expression for W in terms of P and change in V? What is the W in the first step? (hint: it involves an integral involving P and V and you have to substitute for P in terms of V and T in that integral) Is there any work done by the gas in the second step (isochoric)?

-What is the change in entropy?

Write out the expression for $\Delta S$ (hint: it involves Q and T and an integral)

What is the heat flow in the first step? What is the temperature? What is $\Delta S$?

What is the heat flow in the second step? What is the relationship between dQ and dT? Put that into the integral and integrate.

AM