Thermodynamic second derivatives?

  • Thread starter maistral
  • Start date
This is for research purposes.

I am aware that first derivatives in thermodynamics always occur (a no-brainer). Do second derivatives occur in thermodynamics commonly as well?
 

DrDu

Science Advisor
5,965
721
Yes, they are important to analyse stability, i.e. entropy should be maximal. In statistical thermodynamics, second derivatives of free energy gives you the mean fluctuations of e.g. energy or particle numbers.
 
Hi, thanks for replying. Am i correct to assume that this is d2Q/dT2?

Also, could I ask a reference for this information? Thank you very much!
 
I mean, I need the reference for the writeup. Thank you!
 

Mapes

Science Advisor
Homework Helper
Gold Member
2,592
17
This is for research purposes.

I am aware that first derivatives in thermodynamics always occur (a no-brainer). Do second derivatives occur in thermodynamics commonly as well?
And how! Material properties are second derivatives of thermodynamic potentials. For example, the thermal expansion coefficient is $$\alpha_V=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)=\frac{1}{V}\left(\frac{\partial^2 G}{\partial T\partial P}\right)$$ The stiffness is $$E=\left(\frac{\partial\sigma}{\partial\epsilon}\right)=\frac{1}{V}\left(\frac{\partial^2 U}{\partial\epsilon^2}\right)$$ The heat capacity is $$c=T\left(\frac{\partial S}{\partial T}\right)=-T\left(\frac{\partial^2 G}{\partial T^2}\right)$$ And so on.
 
Last edited:
D

DoItForYourself

In the book "Thermodynamics foundations and applications" (E. P. Gyftopoulos, G. P. Beretta), Chapters 9 and 10 they often use the second derivative of entropy.
 
19,033
3,717
And how! Material properties are second derivatives of thermodynamic potentials. For example, the thermal expansion coefficient is $$\alpha_V=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)=\frac{1}{V}\left(\frac{\partial^2 G}{\partial T\partial P}\right)$$ The stiffness is $$E=\left(\frac{\partial\sigma}{\partial\epsilon}\right)=\frac{1}{V}\left(\frac{\partial^2 U}{\partial\epsilon^2}\right)$$ The heat capacity is $$c=T\left(\frac{\partial S}{\partial T}\right)=T\left(\frac{\partial^2 G}{\partial T^2}\right)$$ And so on.
Regarding the last equation, should there be a minus sign? dG=-SdT+VdP
 

Mapes

Science Advisor
Homework Helper
Gold Member
2,592
17
As always, thank you Chester! Edited to fix.

And the reason I should have caught that is that the curves of the Gibbs free energy have an increasingly negative slope with increasing temperature. And when drawn correctly, they end up at T = 0 K as a straight flat line, because the entropy and the heat capacity are zero at absolute zero.
 
In the book "Thermodynamics foundations and applications" (E. P. Gyftopoulos, G. P. Beretta), Chapters 9 and 10 they often use the second derivative of entropy.
Thank you very much. I'll try and get the resource; this will be of great importance to my study :biggrin:

For now, I'm relaxing and playing around with Laplace transforms. Thanks again!
 

Want to reply to this thread?

"Thermodynamic second derivatives?" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top