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Checking my reasoning on a carnot cycle problem

  1. Mar 16, 2014 #1
    1. The problem statement, all variables and given/known data

    An approximate equation of state for a gas is [itex]P(v-b) = RT[/itex] with b constant. The specific internal energy of the gas is [itex] u = c_vT + const [/itex]. a) show that the specific heat at constant pressure of this gas is [itex] c_v +R[/itex] and b) show that the equation of a reversible adabiatic process is [itex] P(v-b)^{\gamma} = const [/itex] and c) show that the efficiency of a carnot cycle using this gas as a working substance is the same as tht for an ideal gas assuming [itex] \left(\frac{\partial u}{\partial v}\right)_T = 0[/itex]

    2. The attempt at a solution

    I did the following for part a: we know that [itex] c_P - c_V = \left[ \left( \frac{\partial u}{\partial v} \right)_T + P \right] \left( \frac{\partial v}{\partial T} \right)_P [/itex] so since [itex] \left( \frac{\partial v}{\partial T} \right)_P = \frac{R}{P} [/itex] and [itex]\left( \frac{\partial u}{\partial v} \right)_T = 0 [/itex] we plug all that into the first equation and we end up with [itex]c_P = R + c_V [/itex]

    For part b) we know that in any reversible adabiatic process

    [itex] \left( \frac{\partial P}{\partial v}\right)_s = \frac{c_P}{c_V} \left( \frac{\partial P}{\partial v} \right)_T [/itex]

    which means, since [itex] \frac{c_P}{c_V} = \gamma [/itex] and for an ideal gas [itex]\frac{dP}{P} + \gamma \frac{dv}{v} [/itex] then [itex] ln P + \gamma ln v = 0[/itex] and we end up with [itex]Pv^{\gamma} [/itex] which when we plug in our original expressions from the equation of state we have [itex]\frac{RT}{(v-b)}\left(\frac{RT}{P} + b\right)^{\gamma}= const [/itex] and that gets us [itex]P(v-b)^{\gamma} [/itex] with a little algebra.

    So far so good, but the last part of the problem stumps me a bit and I think there's some really simple thing I am missing.

    Here's the thing: I know the efficiency of a Carnot cycle is [itex]\eta = \frac{T_2 - T_1}{T_2} [/itex] and [itex]\eta = \frac{Q_2 - Q_1}{Q_2} [/itex] also. If I just plugged in T1 and T2 into the equation of state I suppose I might get an answer, and I wondered if I should just do that and the problem really is that simple. Or, given that they are asking us to assume something about the internal energy of the gas that there's some other silly step I should do. I know that this isn't that complicated so maybe I am just overthinking it. By inspection I suspect that doing the plug-in-T1 method I will get the same answer as for an ideal gas, since [itex]T = \frac{P(v-b)}{R}[/itex]. But I wasn't sure if I should have a v1 or a P1 in there -- I was sure one would be left constant but I wasn't sure which one.

    Anyhow, any help is appreciated.
     
    Last edited: Mar 16, 2014
  2. jcsd
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