Thermal efficiency of a heat engine

[Mangoes]

Homework Statement

A possible ideal-gas cycle operates as follows:
(i) from an initial state (p1,V1), the gas is cooled at constant pressure to (p1,V2)
(ii) the gas is heated at constant volume to (p2,V2)
(iii) the gas expands adiabatically back to (p1,V1).
Assuming constant heat capacities, show that thermal efficiency is given by:

[tex] 1 - γ\frac{(V_1/V_2) - 1}{(p_2/p_1) - 1} [/tex]

2. Homework Equations

For adiabatic processes:
[tex] PV^γ = constant [/tex]

The Attempt at a Solution

For a cyclic process, ΔU = 0, so denoting Q_h as heat coming in system and Q_l as heat leaving system,

[tex] W = Q_h - Q_l [/tex]

Thermal efficiency is defined as

[tex] η = W/Q_h = 1 - \frac{Q_l}{Q_h} [/tex]

(i) has heat flowing out of the system, (ii) has heat flowing in the system, (iii) is adiabatic so heat is zero.

For (i), p is constant and I assume ideal gas

[tex] W = -p(ΔV) = -nR(T_2 - T_1) [/tex]

[tex] ΔU = nC_VΔT [/tex]

By first law,

[tex] Q_l = nC_VΔT + nR(T_2 - T_1) = n(R + C_V)(T_2 - T_1) = nC_p(T_2 - T_1) [/tex]

For (ii) V is constant so work must be zero. That means change in internal energy is equal to heat gained,

[tex] Q_h = nC_V(T_3 - T_2) [/tex]

Here's where I'm getting stuck. If I stick this in into my definition of thermal efficiency,

[tex] η = 1 - \frac{C_p(T_2 - T_1)}{C_V(T_3 - T_2)} [/tex]

I'm aware that since (iii) is adiabatic, it is true that

[tex] T_3V^{γ-1}_2 = T_1V^{γ-1}_1 [/tex]

I've tried using the above to write an expression for T₃ to eliminate it in my expression for thermal efficiency, but it ends up being a huge mess and I don't see how I can take out gamma from the exponent into a multiplying factor as is seen in the result I'm supposed to get to, leading me to think one of my steps is wrong. Would appreciate any help/insight in this.

EDIT: Fixed a step where I mixed U - W with W - U. Had a negative sign that didn't belong.

 

[TSny]

[tex] η = 1 - \frac{C_p(T_2 - T_1)}{C_V(T_3 - T_2)} [/tex]

Good so far. Don't forget the ideal gas law.

 

[Mangoes]

Well, my goal is pretty much getting from

[tex] \frac{T_2 - T_1}{T_3 - T_2} [/tex]

to

[tex] \frac{(V_1/V_2) - 1}{(P_2/P_1) - 1} [/tex]

Applying ideal gas law would make my expression in terms of different Δ(PV). I'm not sure if there's something obvious with the algebra I'm missing. I'm not really seeing how I can express T₃ in terms of either the second or first state.

 

[TSny]

[tex] \frac{T_2 - T_1}{T_3 - T_2} [/tex]

Divide numerator and denominator by T₂.