(a) Calculate the specific heat at constant volume of water vapor, assuming the nonlinear triatomic molecule has three translational and three rotational degrees of freedom and that vibrational motion does not contribute. The molar mass of water is 18.0 g/ mol. (b) The actual specific heat of water vapor at low pressures is about 2000 J / kg ? K. Compare this with your calculation and on the actual role of vibrational motion.
Solution 44E Step 1 The thermal energy per degrees of freedom is given by 1 E 1 k 2 B So if there are f degrees of freedom per molecule, then the energy per molecule is given by f E 2 k T B 2 Now if there are N number of molecule in the gas, then the total thermal energy is given by E = Nk T 2 B From the above equation we can write that f E = Nk2T B Now, the number of molecule per mole is given by N = 6.02 × 10 23/mol As stated in the problem, the number of degrees of freedom for the water molecule is f = 3 + 3 = 6 Hence the molar specific heat at constant volume will be for water vapor will be c = E = Nk = (6.02 × 10 )(1.38 × 10 23 J/mol.K) = 24.9 J/mol v T 2 B 2 3 Now the molar mass of water is M = 18.0 g/mol = 18.0 × 10 kg/mol So the specific heat at constant volume of the water vapor is C = cv= 24.93/mol = 1385 J/kg V M 18.0×10 kg/mol