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