 # MOLAR SPECIFIC HEAT AT CONSTANT PRESSURE

 MOLAR SPECIFIC HEAT AT CONSTANT PRESSURE “The a mount of heat energy required to raise the temperature of one mole of a gasby one Kelvin at constant pressure is called molar specific heat at constant pressure”. It is denoted by Cp. MATHEMATICAL EXPRESSION If DQp is the amount of heat is supplied at constant pressure to ‘n’ moles of a gas to increase the   temperature by DT K, then DQp = n CpDT OR Cp = D Qp /n D T heat supplied at constant pressure is consumed in two purposes:  (1) To raise the temperature of gas.  (2) To do work against external pressure. SHOW THAT CP – CV = R Consider ‘n’ moles of an ideal gas contained in a cylinder fitted with a frictionless piston. If the piston is   fixed and the gas is heated, its volume remains constant and all the heat supplied goes to increase the   internal energy of the molecules due to which the temperature of the gas increases. If DQv is the amount   of heat supplied and DT is the rise in temperature then, DQv = n CvD T

 The pressure of the gas increases during this process, but no work is donebecause the volume is kept constant.Hence D W = 0. applying first law of thermodynamics Heat supplied = Increase in internal energy + Work done DQv = D U + 0DQv = D UORnCvD T = D U If the piston is free to move, the gas may be allowed to expand at a constant pressure. Let the amount of   heat supplied is now is DQp. The addition of heat causes two changes in the system: Increase in internal energy Work done against external pressure  According to the first law of thermodynamics: DQ = DU + DW               {But DW = PDV} DQP = DU + PDV Since DQp = nCpDT  and    DU = nCvDT , therefore, nCpDT = nCvDT + PDV………………..(1) We know that PV = nRT  At T1 Kelvin: PV1 = nRT1 …..(a)  At T2 Kelvin: PV2 = nRT2…..(b)  Subtracting (a) from (b)  PV2 – PV1= nRT2 – nRT1  P(V2 – V1)= nR(T2 – T1)           {(V2 – V1) = DV and (T2 – T1) = DT }  PDV = nRDT Putting the value of PDV in equation (1) nCpDT = nCvDT + nRDT nCpDT = nDT(Cv + R)Cp = (Cv + R)Cp – Cv = R