Question

The molecule 'H'F has a bond length of 0.9256 Å (9.256 x 10-11 m) and a spring constant, k, of 920 Nm"! The atomic masses are 1.007825 g mol-' for 'H and 18.998403 g mol-' for ''F. (a) Calculate the energy difference between the two lowest rotational states for 'H°F? Express your answer in cm! Calculate the energy difference between the two lowest vibrational states for 'H'F? Express your answer in cm! Assuming that we have a 1 cm x 1 cm x 1 cm (0.01 m x 0.01 m x 0.01 m) cubic container. Calculate the energy difference between the lowest energy translational state (n. = 1, ny = 1, n = - 1) and the one with nx = 2, ny = 1, n = 1? Express your answer in cm. (d) Comment on the energy spacings you found in parts (a), (b), and (c). (c)

Answer 1

* c =CH2 Answer-a Dyruvate Given the HF has a bond length of 0.9256 A i The reduced mass M = mixm2 my + m2 11.007825 6.023 x 1023 18-998403 6.023x 1023 (1007825 + 18.998403) 6.023 x 1023 1.007825 x 18.998403 1007825 + 18.998403 6.023 x 10²3 = 1.589x1024 g Hence, the moment of Inertia (I) = 182 = 1.589x1024 x (0-4256x108)? = 1.36 x 17 40 g.com Hence, the rotational constant (B) = 8 T2cI 6.626 x 1034 8x(3:14) 3x 108 x 1.36 10 40 = 2.059x104 cm u. We know states- the energy of rotational E = h CB T (J+1) e level, n=0 and for and For wwest lowest level n=1, hence, the energy gap = hc Bx (1+1) - hcbxo = 2 x 6.626x1034 x 2.059x10 x 3x108 = 8:186 x 1029 1

Answer-b NA Given the spring Constant k for 'H_PF is 920 Nm'. We know, the vibrational energy level En = (1+1/2) havib where, drib = 2 For lowest level, n=0 and for second lowest x level n= 1 hence, the energy gap = /harib-/halvib = harib 920 6.626x1034 2x 3.14 V 1.589 x 10-24 = 2.54x1022J

COO Answer-c е кию, the translational energy level- Given a=b=0= 1cm a = bi = 0.01 m the E energy for Mx = 1, y = 1 and M2 = 1 l Choose 0.'ovs ] (6620x1080 To 01) (0.01² 0.012 SX 1.599x = 1.036x1039 J the en 22 for, nx=2, My =1, M2=1, the energy level, E como eles] = 204 2160p & Energy gap for these two levels. 4 E=E - Eo = 2:072 x 1039 - 1:036*10 39 y - 1.036 * 1039 1
Answer -d :

The energy spacing in translational energy is lowest, and for vibrational energy this gap is highest.

The order for energy gap is -

Vibrational energy gap > rotational energy gap > translational energy gap.