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Polytropic Process

  1. Mar 25, 2017 #1
    1. The problem statement, all variables and given/known data
    One mole of an ideal gas is reversibly compressed from 1 bar to P2
    bar that results in temperature increase from 400 K to 950 K. Calculate the heat
    transferred during the process and the final pressure, if the path followed by the gas can
    be given by PV1.55 = , and the molar heat capacity of the gas is given by

    Cp/R = 3.85 + 0.57×10-3T (where T is in K)

    2. Relevant equations
    dU=dQ+dW

    3. The attempt at a solution
    my attempt at a solution.

    1. I used the first law of thermodynamics to get

    dU=dQ+dW

    solving for dQ

    dQ=dU-dW ===> dQ=CvdT+PdV

    2. Using Ideal gas solved for T then differentiated

    RdT=PdV +VdP

    3. It was given that PV1.55=Constant

    I know since it is a real number I can use steps of a polytropic process

    PVδ=Constant


    This is where I am getting hung up. My books steps to solution say that using


    RdT=PdV +VdP
    solving for PdV

    is the next step to solve the above expression PVδ=Constant.

    It then says that PδVδ-1dV=-VδdP from which VdP=-PδdV

    My question is how did they derive that expression from the PVδ=Constant

    the end result being that

    PdV=RdT/(1-δ)


    Any and all help clarifying this for me would be greatly appreciated I have other formula that are all plug and chug but I would rather understand the steps involved in solving the problem with these constants.

    Best Regards,

    D


     
    Last edited: Mar 25, 2017
  2. jcsd
  3. Mar 25, 2017 #2

    vela

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    What do you mean by "solve the above expression"? Solve it for what?

    Hint: Consider the multivariable function ##f(p,V) = \text{constant}##. If you differentiate it, you get
    $$df = \frac{\partial f}{\partial p}dP + \frac{\partial f}{\partial V}dV = 0.$$
     
  4. Mar 26, 2017 #3


    I am trying to use that PVδ=constant

    The book uses that to get the term of work into a term where the n factor can be used.

    Here is a picture of part of the solution I am referring to and it has the part in question highlighted

    problem.png


    My question is how are they coming to that answer for work from that expression? Is it from your hint?
     
  5. Mar 26, 2017 #4

    This is what I was looking for thank you!
     
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