- #1

tjackson3

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## Homework Statement

A possible ideal gas cycle operates as follows:

(i) From an initial state [tex](p_1,V_1)[/tex], the gas is cooled at constant pressure to [tex](p_1,V_2)[/tex];

(ii) the gas is heated at constant volume to [tex](p_2,V_2)[/tex];

(iii) the gas expands adiabatically back to [tex](p_1,V_1)[/tex].

Assuming constant heat capacities, show that the thermal efficiency is

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

where [tex]\gamma = c_p/c_v[/tex]

## Homework Equations

Carnot efficiency: [tex]\nu = \frac{W}{Q_H} = 1 - \frac{T_l}{T_h}[/tex]

In an adiabatic process, [tex]pV^{\gamma},TV^{\gamma-1},p^{1-\gamma}T^{\gamma}[/tex] are all constant.

## The Attempt at a Solution

I've been spinning my wheels a lot with this one, and I think the issue may be algebraic. My first thought was that I'd calculate the work done in the cycle. For part (i), it's just [tex]p(V_1-V_2)[/tex]. For part (ii), it's zero, since dV = 0. For part (iii), I used the fact that since it's adiabatic, [tex]pV^{\gamma}[/tex] is constant, which I'll call k. Then the work becomes

[tex]W_3 = \frac{k}{\gamma-1}(V_2^{1-\gamma} - V_1^{1-\gamma})[/tex]

You can set [tex]k = p_1V_1^{\gamma} = p_2V_2^{\gamma}[/tex], but there doesn't seem to be a preferable way. Adding these together gives you

[tex]W = p(V_1-V_2) + \frac{p_iV_i^{\gamma}}{\gamma-1}(V_2^{1-\gamma} - V_1^{1-\gamma})[/tex]

Since none of these processes are isothermal, I can't figure out a meaningful expression for [tex]Q_H[/tex], so I'm stuck here. Any help is very much appreciated!