Thermodynamics-internal energy of a surface of a liquid

[rayman123]

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Homework Statement

The differential of the internal energy of a surface of a liquid with surface tension \gamma and area A may be written as
dU=TdS+\gamma dA

Write down the corresponding form of the Helmholtz free energy F= U -TS. Using the fact that these equations involve exact differentials,derive the Maxwell relation
(\frac{\partial S}{\partial A})_{T}=-(\frac{\partial \gamma}{\partial T})_{A}
The internal energy and the entropy are proportional to the area A. Show that the internal energy per unit area is
u(T)= \frac{U}{A}=\gamma-T-(\frac{\partial \gamma}{\partial T})_{A}

Homework Equations

The first part I solved in this way:

F= U-TS
dF= dU-Tds-Sdt
dF= TdS+dW-Tds-Sdt=dW-Sdt
I found on the internet that the mechanical work needed to increase a surface is dW=\gamma dA
so my dF= -SdT+\gamma dA

How to show the last expresion?
Anyone willing to help?

 

[Thaakisfox]

For the maxwell relation use the differential form of the Helmholtz free energy you derived, and read off the partial derivatives, now differentiate them again and use young's theorem.

 

[rayman123]

hm I am not quite following...could you give me an example?

 

[Thaakisfox]

dF=-SdT + \gamma dA

this is a differential for the function F(T,A). Hence we can see that:

\frac{\partial F}{\partial T}\bigg |_A = -S

Now differentiate wrt A keeping T constant:

\frac{\partial^2 F}{\partial A\partial T} = -\frac{\partial S}{\partial A}\bigg |_T

Now do the same for the other part:

\frac{\partial F}{\partial A}\bigg |_T = \gamma

differentiate wrt. T keeping A constant and then use Young's theorem, and you are done

 

[rayman123]

oh that makes really sense

is this correct

\frac{\partial^2 F}{\partial T\partial A} = (\frac{\partial \gamma}{\partial T})_A

so (\frac{\partial S}{\partial A})_{T}=-(\frac{\partial \gamma}{\partial T})_{A} by Young's theorem.

and the last part

u(T)= \frac{U}{A}= \frac{TS+\gamma A}{A}=\frac{TS}{A}+\gamma= \gamma -T(\frac{\partial \gamma}{\partial T})_{A}[/itex]<br /> is this correct?