Thermodynamics: derivation for q @ constant P

[iScience]

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?

 

[Philip Wood]

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?

 

[iScience]

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?

 

[stevendaryl]

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?

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 $dX$ for some function of state $X$. $dq$ is not an exact differential, because $q$ 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, $dW$, 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.

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

The only nonexact differentials that I know of that have names are work and heat:
$dW = -P dV$ and $dq = T dS$

 

[Chestermiller]

dq is equal to TdS only for a reversible process.

Chet