For a certain system, the thermodynamic energy U is given as a function of S,V,and n by

U = U(S,V,n) = Kn^5/3V^−2/3e^2S/3nR,

where S is the entropy, V is the volume, n is the number of moles, K is a constant, and R is the gas constant.
(a) According to thermodynamic theory, T = (∂U/∂S)_{V ,n}. Find an expression for (∂U/∂S)_{V ,n}.
(b) According to thermodynamic theory, the pressure P = − (∂U/∂V )_S,n. Find an expression for (∂U/∂V )_S,n.
(c) Find an expression for (∂U/∂n)_S,V.
(d) Find dU in terms of dS,dV, and dn.