please answer all parts with step by step

2-1. Thermal Conductivity and the Second Law Consider heat conduction in a stationary material (v=0). The objective is to show that the second law of thermodynamics requires that the thermal conductivity (k) be positive. [Actually, as described in Von Baeyer (1998), the observation that heat always flows from hotter to colder objects was some of the earliest evidence in favor of the second law.] (a) State the differential conservation equation for the internal energy per unit mass, U^. (b) Derive the conservation equation for entropy per unit mass, S^, assuming that is constant and HV=0. (Hint: Begin with the equation for U^ and then use the relationship between U^ and S^.) (c) Suppose that it is desired to write the result of part (b) in the form of Eq. (2.3-7), with the entropy flux denoted by js and the volumetric rate of entropy production given by . If the entropy flux is defined as js=q/T, determine . d) According to the second law, the total entropy of a system undergoing a spontaneous change must increase with time. Thus, for a fixed control volume, dtdVS^dV>0. Using the results of part (c), show that this requires that k>0. Consider a system that is nonisothermal but adiabatic