Clarification on derivations of specific heats in fluids

Click For Summary

Discussion Overview

The discussion revolves around the derivations of specific heats in fluids, specifically focusing on the definitions and relationships involving internal energy, enthalpy, and the first law of thermodynamics. Participants explore the mathematical expressions and conceptual understanding of these thermodynamic properties.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents their understanding of specific heats and seeks clarification on the derivation of CP, questioning the treatment of the enthalpy definition and its implications for the equations used.
  • Another participant challenges the initial definition of enthalpy provided, asserting that enthalpy is a state function defined as H = U + pV, and provides a corrected derivation for dH.
  • A later reply acknowledges a typo in the initial post and revisits the derivation, ultimately aligning with the corrected definition of enthalpy and clarifying the relationship between the terms involved.
  • One participant expresses agreement with the clarification provided in the discussion.

Areas of Agreement / Disagreement

There is disagreement regarding the initial definition of enthalpy, with some participants asserting it is incorrect while others accept the corrections made. The discussion remains unresolved in terms of the initial participant's understanding of the derivation process.

Contextual Notes

The discussion highlights potential misunderstandings related to the definitions and relationships in thermodynamics, particularly concerning the treatment of state functions and the application of the first law of thermodynamics. There are unresolved aspects regarding the clarity of the derivation steps and the implications of the definitions used.

Kori Smith
Messages
3
Reaction score
0
Hello! I understand what specific heats are and how to derive them. I just feel that I'm missing a little something in the methodology.

Consider the 1st law of thermodynamics and the definition of enthalpy:

1) dU = δQ -δW = δQ - PdV
2) H = Q - VP

For the derivation of CV, dV = 0 and the relationship becomes

(∂U/∂T)V = (∂Q/∂T)V = CV

For the derivation of CP, something happens that I don't quite understand. Sources I've found say that the incremental form of the enthalpy relation is given as

3) dH = δQ - VdP

since dP = 0, it becomes

(∂U/∂T)P = (∂Q/∂T)P = CP

but why do we write it like this? Wouldn't the chain rule for differentiation apply to d(VP) s.t. it becomes

d(VP) = VdP + PdV

in which case, where does the PdV component in EQ (3) go? Thanks ahead of time for the clarification!
 
Science news on Phys.org
The definition of enthalpy is not correct. Enthalpy is a state function, like U, T, V or p. It is defined by:
H = U + pV. Then instead of your equ. (3):

dH = dU + pdV + Vdp = δQ + Vdp, so that, for a constant pressure process, dp = 0, and
dH = δQ,
(∂H/∂T)p = (∂Q/∂T)p = Cp

The problem with the definition of enthalpy that you gave is that there is no state function called Q, so you cannot define any state function in terms of something called Q.
 
Chandra Prayaga said:
The definition of enthalpy is not correct. Enthalpy is a state function, like U, T, V or p. It is defined by:
H = U + pV. Then instead of your equ. (3):

dH = dU + pdV + Vdp = δQ + Vdp, so that, for a constant pressure process, dp = 0, and
dH = δQ,
(∂H/∂T)p = (∂Q/∂T)p = Cp

The problem with the definition of enthalpy that you gave is that there is no state function called Q, so you cannot define any state function in terms of something called Q.

Woops! I made a typo. As usual, my issue boils down to simple errors. When I look at it again, I see that

dH = dU + pdV + Vdp = δQ - δW +pdV + Vdp = δQ + Vdp + (pdV - δW) = δQ + Vdp

since δW = pdV (pressure-volume work). Which is exactly as it should be and explains where the missing pdV went.
 
Absolutely.
 

Similar threads

Undergrad Graphical difference between adiabatic and isothermal processes.
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Change of Variables in Thermodynamics
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Question regarding dh, du, cp, cv for ideal gases
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Question about the first law of thermodynamics
  • · Replies 3 ·
Replies
3
Views
2K
Chemistry How to derive 2nd law extremum principles for U, A, G, and H?
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Why Do Heat Capacities Use Derivatives of Entropy in Their Formulas?
  • · Replies 5 ·
Replies
5
Views
9K
Undergrad Derivation of ideal gas heat capacity relationship
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Temperature change (first law of thermodynamics)
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Proof of fundamental thermodynamics equation for open systems
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Molar Specific Heat (gas) at varying pressure and volume?
  • · Replies 5 ·
Replies
5
Views
2K