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## Homework Statement

A hypothetical engine, with an ideal gas as the working substance, operates on the cycle shown in Figure 1. Show that the efficiency of this engine is

[tex] e = 1 - \frac{1}{\gamma}\left(\frac{1 - \frac{p_3}{p_1}}{1 - \frac{v_3}{v_1}}\right) [/tex]

Where [tex] \gamma = \frac{c_p}{c_v} [/tex]

## Homework Equations

Efficiency = Benefit/Cost

Benefit = Work out

Cost = Heat in

PV = nRt

[tex] \Delta U = Q_{hot} - Q_{cold} - Work [/tex]

Work = pdv

## The Attempt at a Solution

The Graph is attached, but basically it has a Adiabatic curve, which at the bottom goes up vertically, then left horizontally, back to make a cycle.

So far I have:

E = Work/Cost

Work = pdv

Cost = Q

[tex] w = pdv [/tex]

Since p is not constant, use ideal law,

[tex] w = \frac{nRt}{v}dv [/tex]

thus

[tex] w = \frac{nRt}{v}dv [/tex]

[tex] w = nRt [ln(v)]^{v_1}_{v_2} [/tex]

I also have:

[tex] e = 1 - \frac{Q_{cold}}{Q_{hot}}[/tex],

from

[tex] \Delta U = Q_{hot} - Q_{cold} - Work [/tex]

Which is cylci and this delta u = 0.

I am going the ruight way about this problem?

Many thanks,

TFM