Question

Calculate the zero point energies for ¹H¹⁹F and ²D¹⁹F. Compare the difference in the zero point energies to k_BT at 298 K.

Answer 1

Force constant of 1H19F = 970 N/m

Frequency, v = 1/2

=  1/2[ 970 / (1.008 amu * 18.998 amu / 1.008 + 18.998) * 1.661x10^-27 kg/amu ]

= 1/2[ 970 / (19.149984 / 20.006) * 1.661x10^-27 kg ]

= 1/2[ 970 / (1.5899x10^-27 kg ]

= 1.243x10^14 s-1

Zero point energy of 1H19F,

E_o = hv / 2 = 6.626x10^-34 * 1.243x10^14 / 2

E_o = 4.11x10^-20 J

For 2D19F -

Frequency, v = 1/2

=  1/2[ 970 / (2.0140 amu * 18.998 amu / 2.0140 + 18.998) * 1.661x10^-27 kg/amu ]

= 1/2[ 970 / (38.2619 / 21.012) * 1.661x10^-27 kg ]

= 1/2[ 970 / (3.0246x10^-27 kg ]

= 9.013x10^13 s-1

Zero point energy of 2D19F,

E_o = hv / 2 = 6.626x10^-34 * 9.013x10^13 / 2

E_o = 2.98x10^-20 J

Difference in zero point energies to kBT at 298 K -

E_o^HF - E_o^DF / kT = (4.11x10^-20 - 2.98x10^-20) / 298 * 1.38x10^-23

= 2.747