An elastomer enthalpy K and an elastomer Gibbs free energy J

An \"elastomer enthalpy\" K and an \"elastomer Gibbs free energy\" J can be defined as follows: K = U - fL J = K - TS = F - fL Using these definitions, obtain equations for the differentials dU, dK, dF and dJ which are analogous to the classical thermodynamic relationships obtained when P, V and T are the state variables. Write down the Maxwell relationships obtained from the equations in (a). Derive the two general expressions for the elastomer heat capacity C below. From these, write expressions for the heat capacity at constant length C_L and the heat capacity at constant force C_F. C = (partial differential U/partial differential T)_L + [(partial differential U/partial differential L)_T - f]dL/dT C = (partial differential K/partial differential)_f + [(partial differential k/partial differential f)_T + L]dt/dT

Solution

Taking diferentials analogous to differentials of U H S etc when T constant, here we put f constant

dK= dU-fdL

dJ =dF-fdL = dK -TdS (analogous to dG =dH-TdS)

Unfortunately you have no explained the meanings of the terms f, L so I cannot help further.

You can substitute the analgous relations in the MW derivations.

Infinitesimally , dJ = dF -fdL -Ldf, = dF-fdL-Ldf

b) Maxwell relations:

dU = TdS -PdV

dH= TdS +VdP

dG = -SdT+VdP

from the analogous eqns for Maxwells relations ie

dT/dV = -dP/dS etc

derive the same fro the above.

c) Cp =dH/dT p const

Cv = dE/dT v const.

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