A mass m of a material is heated reversibly from temperature T1 to T2 at a constant pressure of P. In this problem, we are going to determine an expression for the change in entropy, ΔS. Assume we can heat the given material infinitesimally slowly so that the process is reversible. Therefore, heat in any infinitesimal step is given by the following:
dQ = mc_p dT,
where c_p is the specific heat. In general, the heat capacity of a material can be temperature-dependent. A general temperature-dependent empirical form for the specific heat for ideal gases and incompressible liquids such as water is as follows:
c_p(T) = A + BT + CT² + DT⁻²,
where A, B, C, and D are substance-dependent constants that can be determined empirically.
(a) Write an expression for the change in entropy ΔS for the system assuming the empirically temperature-dependent form of the specific heat.
(b) Calculate the change in entropy ın call for a sample of water with mass rn = 2.2 kg and changing temperature from T1 = 21.4°C to T2 = (21.4+10)°C The specific heat c of water is 1,000 cal /kg/K.