Thermodynamic second derivatives?

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Second derivatives in thermodynamics are crucial for analyzing stability, particularly regarding the maximization of entropy. They are also essential in statistical thermodynamics, where they relate to mean fluctuations of energy or particle numbers. Material properties, such as thermal expansion coefficients and heat capacity, can be expressed as second derivatives of thermodynamic potentials. Key references for this information include "Thermodynamics Foundations and Applications" by E. P. Gyftopoulos and G. P. Beretta, particularly Chapters 9 and 10. Understanding these concepts is vital for research in thermodynamics.
maistral
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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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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!
 
maistral said:
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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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.
 
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Mapes said:
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
 
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!
 
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DoItForYourself said:
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!
 
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