Efficiency in a thermodynamic cycle

[Alexandre Ricardo]

I have to calculate the efficiency of the above cycle for an ideal gas, that goes at the order a-b-c-a:

I know that the answer is 16/97 but I can't calculate the heat in the a-b stage.

 

[Chestermiller]

I have to calculate the efficiency of the above cycle for an ideal gas, that goes at the order a-b-c-a:
View attachment 225655
I know that the answer is 16/97 but I can't calculate the heat in the a-b stage.

Let's see what you have done in detail for the a-b stage.

 

[Alexandre Ricardo]

Let's see what you have done in detail for the a-b stage.

Right. I started using the Gibbs equation:
$TdS=PdV+dU$
Using the relation:
∂U/∂v|_T=T∂P/∂T|_v-P

And making some manipulations I obtained:

dQ_a-b=c_vdT+PdV

Considering an ideal gas. Using the ideal gas state equation, ΔT=0 and PdV=W_a-b.

And I found ΔQ=W_a-b=3P₀V₀/4

But i think there is something wrong in my calculations and I don't know if I would consider ΔT=0 (even if dT≠0)

 

[Chestermiller]

Right. I started using the Gibbs equation:
$TdS=PdV+dU$
Using the relation:
∂U/∂v|_T=T∂P/∂T|_v-P

And making some manipulations I obtained:

dQ_a-b=c_vdT+PdV

Considering an ideal gas. Using the ideal gas state equation, ΔT=0 and PdV=W_a-b.

And I found ΔQ=W_a-b=3P₀V₀/4

But i think there is something wrong in my calculations and I don't know if I would consider ΔT=0 (even if dT≠0)

Calculate the variation of Q along the path a-b, and see how it varies. Does it change sign along the way?