Question

The force constant for CO is 1860 N/m-1. determine the value of the zero point energy of CO in Joules/molecule and kJ/mole. Show why the answer is 12.84 kJ/mole

Answer 1

Zero pont energy per mole E = Nhv/2, N = avagadro's number = 6.023 x 10²³ mol^-1

                       v = (1/2π) (k/M)^1/2

Hence,   E = (Nh/2) [ (1/2π) (k/M)^1/2 ]    -- Eq(1)

                   h = planck's constant = 6.626 x 10^-34 J.s

                  k = force constant = 1860 Nm^-1

mass of carbon = 12 amu   = 12 x 1.66x 10^-27 kg

mass of oxygen = 16 amu   = 16 x 1.66x 10^-27 kg

      M = effective mass of CO = (mass of C. mass of O ) / (mass of C + mass of O)

                             = (12 x 1.66x 10^-27 kg . 16 x 1.66x 10^-27 ) / (12 x 1.66x 10^-27 + 16 x1.66x 10^-27 kg )

                             = 11.4 x 10^-27 kg

Substitute all these values in Eq (1) ,

E = (Nh/2) [ (1/2π) (k/M)^1/2 ]    -- Eq(1)

= [ 6.023 x 10²³ mol^-1 x6.626 x 10^-34 J.s / 2 ] [ (1/2π) (1860 Nm^-1 /11.4 x 10^-27 kg )^1/2 ]

= 128.4 x 10² J/mol

    = 12.84 x 10³ J/mol

= 12.84 kJ/mol

E = 12.84 kJ/mol

Therefore,

Zero point energy of CO = 12.84 kJ/mol