Thermodynamic second derivatives?

[maistral]

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]

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.

 

[maistral]

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!

 

[maistral]

I mean, I need the reference for the writeup. Thank you!

 

[Mapes]

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.

 

[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.

 

[Chestermiller]

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]

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.

 

[maistral]

Thanks guys!

 

[maistral]

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

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