How to Prove Thermal Efficiency for an Imaginary Ideal Gas Cycle?

[jrklx250s]

Homework Statement

Given an imaginary ideal-gas cycle. Assuming constant heat capacities, show that the thermal efficiency is

η = 1 - γ[((V1/V2)-1)/((P3/P2)-1)]

Since i can't show you the cycle we are shown that

l Qh l = which is absolute value of the heat at high temperature = Cv(T3-T2)
l QL l = which is absolute value of the heat at low temperature = Cp(T1-T2)
Cp/Cv = γ

The Attempt at a Solution

Ok so subing in these equations for thermal efficiency
which is

η = 1 - l QL l / l Qh l

we get...

η = 1 - γ(T1 - T2)/(T3 - T2)

η = 1 - γ((T1/T3) - 1)

This imaginary cycle only has a power stroke and we are assuming that its adiabatic...from this we concluded that

T1V1^(γ-1) = T3V2^(γ-1)
T1P2^((1-γ)/y)=T3P3^((1-γ)/y)

divide each equation we get

V1^(γ-1)/P2^((1-γ)/y) = V2^(γ-1)/P3^((1-γ)/y)

Now I am not sure how to rearrange from here to make T1/T3 = (V1/V2)/(P3/P2)

Any suggestions would be greatly appreciative.
Thanks!

 

[jrklx250s]

Since no one has replied I'm assuming some are confused as to what I'm talking about so here is the ideal gas cycle that I need to calculate the thermal efficiency from.
Here is the link to the picture of the cycle
http://imageshack.us/photo/my-images/411/img1048u.jpg/

 

[I like Serena]

Hi jrklx250s!

Did you already try to apply the ideal gas law PV=RT?

 

[jrklx250s]

Hi Serena,

Yes I believe so when i calculated the adiabatic processes for the power stroke... which i concluded that they were

T1V1^(γ-1) = T3V2^(γ-1)
T1P2^((1-γ)/y)=T3P3^((1-γ)/y)

And Since I need to make T1/T3 = (V1/V2)/(P3/P2)

This means that T1 = (V1*P2)

and T3 = (V2*P3)

not sure how to conclude these from the two equations above. And I know its a simple alegbraic rearrangement that I am missing here.

 

[I like Serena]

Careful. Let's start with (V1/V2)/(P3/P2).
With some fraction manipulations this is equal to (P2*V1) / (P3*V2).

Looking at the diagram you posted I can see that P1=P2 and that V1=V3.
Furthermore you have that for instance P1*V1 = R*T1.

Perhaps you can use that?

 

[jrklx250s]

Haha wow...thank you serena I was making this so much more complicated than it was.

Yea of course you can just conclude that since
P1=P2
V2=V3

so therefore...
P1V1=nRT1
P3V3=nRT3

P2V1=nRT1
P3V2=nRT3

solving for both T's

T1=P2V1/nR
T3=P3V2/nR

sub this in my previous equation and we get...
η= 1 - ((V1/V2)-1)/((P3/P2)-1)

Thank you.