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molkee

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## Homework Statement

The energy density (u=E/V) of a thermodynamic system (used as a model of symmetry restoration in the early universe) is given by:

[tex] u(T)=aT^4 + \Lambda(T)[/tex] where [tex] \Lambda =0, T \leq T_0 [/tex] or [tex] \Lambda =\Lambda_0, T>T_0 [/tex]

[tex]k_B T_0 = 10^{14} GeV [/tex]

a) Calculate the Helmholtz free energy for the system

b) Calculate the pressure and entropy from a). To fix any constants of integration, use the condition [tex]p=aT^4/3 [/tex] at *very low *temperatures

c) find the factor by which the volume changes if a container of this stiff is adiabatically and reversibly cooled from T just above T_0 to T just below T_0

d) suppose that the system cools reversible to zero in a metastable phase in which [tex]\Lambda[/tex] remains stuck at [tex]\Lambda_0[/tex] (instead of going to zero below T_0). What are the values of energy and entropy in this limit?

e) the system then spontaneously and rapidly decays to the stable state in which [tex]\Lambda=0[/tex]. Find the final temperature of the system and the entropy change of the transition.

## Homework Equations

will be used in the next section

## The Attempt at a Solution

a) The equation [tex]\left (\frac{\partial (F/T)}{\partial T}\right )_V=-\frac{E}{T^2}[/tex] should be used. After the integration, F/T is defined up to a constant F_0.

b) F(T,V) from a) should be differentiated with respect to T and V.

c) The equation [tex]dE(T,V)+p(T)dV=0[/tex] for adiabatic process should be used. It will give us the connection between T and V in this process.

We know p(T) (from b)) and E(T,V).

d)no idea

e)no idea

Am I doing something wrong or not?

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