Thermodynamics: derivation for q @ constant P

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Discussion Overview

The discussion revolves around the derivation of the heat transfer equation \( q \) at constant pressure in thermodynamics, specifically examining the relationship between heat, entropy, and temperature. It includes theoretical aspects and clarifications regarding the use of differentials in thermodynamic equations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents the equation \( dq = d(ST) = SdT + TdS \) and derives \( \Delta S \) at constant pressure in terms of heat capacity \( C_p \), questioning why \( T \) is kept constant on the \( TdS \) side.
  • Another participant expresses confusion regarding the initial equation and requests justification for it, suggesting that the lack of replies may stem from similar bafflement among others.
  • A participant reflects on their classroom experience where \( dq = TdS \) was used for heat, noting that they assumed \( dq \) was the derivative of \( ST \) and questioned the nature of the term \( SdT \).
  • Further clarification is provided that \( SdT \) is an inexact differential and that \( dq \) is not an exact differential because heat is not a state function, contrasting it with state functions like temperature and entropy.
  • It is noted that \( dq = TdS \) holds true only for reversible processes, highlighting a condition under which the equation applies.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and confusion regarding the equations and concepts presented. There is no consensus on the interpretation of the initial equation or the role of \( SdT \), indicating ongoing debate and uncertainty.

Contextual Notes

The discussion highlights the distinction between exact and inexact differentials in thermodynamics, but does not resolve the assumptions or implications of these distinctions in the context of the equations presented.

iScience
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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?
 
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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?
 
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?
 
iScience said:
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
 
Last edited:
dq is equal to TdS only for a reversible process.

Chet
 

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