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Friday, December 14, 2012

Show that for an ideal gas, Cp - Cv= R


From the definitions , it is clear that two heat capacities are not equal and CP is greater than CV by a factor which is related to the work done. At a constant pressure part of heat absorbed by the system is used up in increasing the internal energy of the system and the other for doing work by the system. While at constant volume the whole of heat absorbed is utilized in increasing the temperature of the system as there is no work done by the system. Thus increase in temperature of the system would be lesser at constant pressure than at constant volume. Thus CP is greater than Cv.
We know             Cp = dH/dT                               …(1)
And                      Cv = dE/dT                               ….(2)
By definition,  H = E + PV for 1 mole of an ideal gas
Or                           H = E + RT                        (PV= RT)
Differentiating w.r.t. temperature, T, we get
                                dH/dT = dE/dT + R
or                            Cp = Cv+ R                            By using (1) and (2)
or                            Cp – Cv = R
                                                                (Shown)

11 comments:

  1. please show that using maxwell relation

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  2. This is an incomplete proof. dH/dT and dE/dT are partial derivatives. So you could have (dH/dT)p (dE/dT)p --or-- (dH/dT)v (dU/dT)v

    for the approach in our solution, you need to show that (dE/dT)p=(dE/dT)v

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  3. explanation is clear but some not catch that ,so give every reactions breifly . not only thse all organic reactions

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  4. This comment has been removed by the author.

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  5. Sir can you please give a graph for the reaction

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  6. Superb explanation sir
    Thanku so much for your efforts.

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