# 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?

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#### 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

Homework Helper
Gold Member
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.

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#### 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

Mentor
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

Homework Helper
Gold Member
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.

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!

"Thermodynamic second derivatives?"

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