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Thermodynamics: derivation for q @ constant P

  1. Oct 17, 2014 #1
    general equation of q in terms of S,T

    $$q=d(ST)=SdT+TdS$$


    deivation of ΔS at constant pressure(in terms of heat cap C_p:

    $$dq=C_{p}dT=TdS$$

    $$\frac{C_{p}}{T}dT=dS$$

    $$C_{p}ln(T_{f}/T_{i}=ΔS$$

    why do we keep T constant on TdS side?
     
  2. jcsd
  3. Oct 17, 2014 #2

    Philip Wood

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    Gold Member

    Maybe you're not getting replies because others are as baffled as I am by your very first equation. Or maybe I'm just being stupid. Could you please justify your first equation?
     
  4. Oct 18, 2014 #3
    oh, sorry.

    well in my thermo class we used to use dq=TdS for heat. in certain equations though id seethe term "SdT", whenever i see situations like this i assume dq is actually the derivative of ST ie d(ST) and that it was due to some special case that we were able to ignore the SdT term.
    bad assumption i guess.

    but if TdS is dq and SdT does not come from q, then what is SdT?
     
  5. Oct 19, 2014 #4

    stevendaryl

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    I'm not sure what kind of answer you are looking for. S dT is an inexact differential thermodynamic quantity with no particular name, as far as I know.

    The difference between an exact differential and an inexact differential is that an exact differential can be written as [itex]dX[/itex] for some function of state [itex]X[/itex]. [itex]dq[/itex] is not an exact differential, because [itex]q[/itex] is not a function of state. The amount of heat you put into a system to end up at a particular state depends on how you get to that state. Similarly, [itex]dW[/itex], the amount of work done on a system, is not an exact differential, either. In contrast, the temperature, the entropy, the pressure, the volume, the internal energy, etc. are functions of state, and so their differentials are exact differentials.

    [itex]T dS[/itex] is not an exact differential, and neither is [itex]S dT[/itex], but the sum is an exact differential: [itex]T dS + S dT = d(ST)[/itex]

    The only nonexact differentials that I know of that have names are work and heat:
    [itex]dW = -P dV[/itex] and [itex]dq = T dS[/itex]
     
    Last edited: Oct 19, 2014
  6. Oct 20, 2014 #5
    dq is equal to TdS only for a reversible process.

    Chet
     
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