Hi everyone! Im a UK grad student working in acoustics. My own background is in EE, so i am largely self taught in thermodynamics. Consequently id really appreciate any insight any of you real physicists can give me with my problem!(adsbygoogle = window.adsbygoogle || []).push({});

Following derivations in books i can derive the enthalpy, H_{A}, of an acoustic medium to be as follows (S is strain, T is stress, D and E are electric filed and electric displacement,B and H are magnetic field, [tex] \theta [/tex] is temperature and [tex] \sigma [/tex] is entropy. Furthermore, the problem is assumed to be adiabatic.

[tex]H_{A} &= U -(ST)-(ED)-(BH)[/tex]

(where [tex]dU = dW + dQ = T_{i}dS_{i} + E_{i}dD_{i} + H_{i}dB_{i} + \theta d\sigma[/tex]), therefore:

[tex]dH_{A} &= - S_{i}dT_{i} - D_{m}dE_{m} - B_{m}dH_{m} + \theta d\sigma [/tex]

so:

[tex]S_{i}=\frac{-\partial{H_{A}}}{\partial{T_{i}}}[/tex]

etc etc for the other non infinitesimal quantities etc.

My problem comes when I expand Si and the other non-state variables from dH_A individually into total differentials ie

[tex] dS_{i} &=(\frac{\partial{S_{i}}}{\partial{T_{i}}})_{E,H,\sigma}dT_{i} + (\frac{\partial{S_{i}}}{\partial{E_{m}}})_{H,\sigma}dE_{m} + (\frac{\partial{S_{i}}}{\partial{H_{m}}})_{H,E}dH_{m} + (\frac{\partial{S_{i}}}{\partial{\sigma}})_{H,E}d\sigma [/tex]

I need to convert this expression into a non-infinitesimal form, ie so the LHS is S and not ds. I know the correct answer from my textbook is:

[tex] S_{i} &=(\frac{\partial{S_{i}}}{\partial{T_{i}}})_{E,H,\sigma}T_{i} + (\frac{\partial{S_{i}}}{\partial{E_{m}}})_{H,\sigma}E_{m} + (\frac{\partial{S_{i}}}{\partial{H_{m}}})_{H,E}H_{m} + (\frac{\partial{S_{i}}}{\partial{\sigma}})_{H,E}d\sigma [/tex]

However I don't see how to obtain this and the book glosses over the details of how to do this! Obviously I need to perform some kind of integration here, but how do I do this?! The thing that is throwing me is that last term involving entropy and temperature. I can see how to integrate this for terms T,E and H ie a term of the form

[tex] (\frac{\partial{S_{i}}}{\partial{T_{i}}})_{E,H,\sigma}dT_{i} [/tex]

converts to [tex] (\frac{\partial{S_{i}}}{\partial{T_{i}}})_{E,H,\sigma}T_{i} [/tex].

Can someone explain to me the significance of [tex](\frac{\partial{S_{i}}}{\partial{\sigma}})_{H,E}d\sigma[/tex] since this term doesn't seem to transform in the same fashion?

Also: Should i ditch the infinitesimal notation for S? At least one book ive seen assumes S is a function of the form [tex] S(T,E,H,\sigma) [/tex] and expands this function using a Maclaurin series: this seems to avoid the tricky infinitesimal quantities. Is this a better approach? Generally, using infinitesimals in calculus seems to be a bit of a minefield, as they are not very rigorous.

Thanks!

-=+Stephen Ellwood+=-

Ultrasound Research Group, Leeds University, UK

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# Thermodyamics, total differentials and integration.

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