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10.9. Consider equilibrium between two phases in a single-component system. Showthat the change in chemical potential with pressure along the phase boundaryis given bydudP phase boundaryA(1/3)A(1/s)where sS/N and sS/V

Question

This problem is (10.9) from a book  "Thermodynamics and Statistical Mechanics An Integrated Approach by M. Scott Shell"

10.9. Consider equilibrium between two phases in a single-component system. Show
that the change in chemical potential with pressure along the phase boundary
is given by
du
dP phase boundary
A(1/3)
A(1/s)
where s
S/N and s
S/V
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10.9. Consider equilibrium between two phases in a single-component system. Show that the change in chemical potential with pressure along the phase boundary is given by du dP phase boundary A(1/3) A(1/s) where s S/N and s S/V

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Step 1

For a single component system in two phases in equilibrium, the relationship between the chemical potential of the two phases is:

μ" (Τ.P ) = μ' (Τ. P)
(1)
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μ" (Τ.P ) = μ' (Τ. P) (1)

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Step 2

Also, the relationship between chemical potential, entropy of the system, and volume of the system is:

.... (2)
du-SdT VdP
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.... (2) du-SdT VdP

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Step 3

Write this equation (2) for both the phases...

du" (T.P) -du" (Т,Р)
-Se dT V dP -s dT+VdP
(e-)dP = (S -s) dT
(S-sP
dP
-ат
AS
dP= dT
AV
Since, S/V
1
- dT
dP
(3)
A(1/5)
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du" (T.P) -du" (Т,Р) -Se dT V dP -s dT+VdP (e-)dP = (S -s) dT (S-sP dP -ат AS dP= dT AV Since, S/V 1 - dT dP (3) A(1/5)

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