The exact differential for the Gibbs energy is given by dG = -SdT + VdP. The form of this differential implies which of the following relationships? JP ӘS (7) - ( ), av V T /aG (º) -(), = P T (37)--(F), P as (*),- (*), = P /aG (*) -- ( ), G др

In the Taylor Expansion, it looks like they've made x_0 = x_1, x_2 and made x = x_1 - a, x_2 - a.  I don't understand how they can use that substitution for x_0 since it is a constant, and x_1 and x_2 are variables that can change.

Calculate the number of microstates that are available in a single atom of carbon in graphite.The standard molar entropy (S°) of carbon is 5.7 J/(mol · K) and the Boltzmann constant (kB) is1.381 × 10 ―23?/?.   HELP PLEASE

the partition function of an ideal gas of diatomic molecules in an external electric field & is [g(V, T, 8)]" Q(N, V, T, 8) N! where (2mmkT 312 (87 IkT -hv/2kT e q(V,T, 8)= V{ h2 (kT' (µ8 sinh kT) h2 (1 – e-hv/kT) Here I is the moment of inertia of the molecule; v is its fundamental vibrational frequency; and u is its dipole moment. Using this partition function along with the thermodynamic relation, dA = -S dT –p dV – M de where M=Nū, where u is the average dipole moment of a molecule in the direction of the external field &, show that kT] coth kT, Sketch this result versus & from & =0 to & =∞ and interpret it.