Question

Calculate the energy separations in joules, kilojoules per mole, and reciprocal centimeters between the levels (a) n = 2 and n = 1, (b) n = 6 and n = 5 of an electron in a box of length 2.0 nm

Answer 1

The energy of a particle in a box is given as follows: 8ml Here, n is the energy state, h is the Planck's constant (6.626x 10-34 Js), m is the mass of electron (9.11x10-5 kg), and I is the length of the box (2x10-ºm). (a) The general formula for energy separation is as follows: AE = (n} –n?)h2 8ml Here, n, is 2 and n is 1. Putting all the values as follows: (2)-(1)) (6.626x10-24 Js) AE = 8(9.11x10-31 kg (2x10 m) = 4.52x10-20 J Therefore, the energy separation in joules is 4.52x 10-20 Jl. The energy separation in kilojoule per moles is given as follows: 11 =10 kJ Therefore, 4.52x10-201 - 4.52x10-20 x 10ºkJ 11 = 4.52x10-23 kJ Converting into kJ per mole as follows: 4.52x10-23 kJ = 4.52x10-23 kJ x6.022x102 mol-1 = 27.22 kJ mol?

Therefore, the energy separation in kilojoule per mole is 27.22 kJ mol-. According to Planck's hypothesis, energy is given as follows: AE = hv Here, h is the Planck's constant, and v is the frequency. Frequency is also given as follows: also Putting these values in energy relation as follows: AE = hcū Putting the values as follows: )(3x10 m ) 4.52x10-20 1 = (6.626x10-34 ū=0.227x10 m- = 0.227x10* cm- Therefore, the energy separation in centimeter inverse is 0.227x10 cm-'.

(b) The general formula for energy separation is as follows: (nz-n?)h2 8m2 Here, n, is 6 and n, is 5. Putting all the values as follows: (6)-(5)?) (6.626x10-24 Js)? AE ='' 8(9.11x10-kg) (2x10 m) =1.66x10-19 J Therefore, the energy separation in joules is 1.66x10-18.

The energy separation in kilojoule per moles is given as follows: 11 =10-kJ Therefore, X10-3 1-1 1.66x 10-19 1 - 1.66 x10-19 =1.66x10-22 kJ Converting into kJ per mole as follows: 1.66x10-19 kJ = 1.66x10-19 kJ x 6.022x102 mol-1 =9.99 x10* kJ mol-1 Therefore, the energy separation in kilojoule per mole is 9.99x104 kJ mol-'. According to Planck's hypothesis, energy is given as follows: AE = hv Here, h is the Planck's constant, and v is the frequency. Frequency is also given as follows: also Putting these values in energy relation as follows: AE = hcū

Putting the values as follows: *)(3x109 m ) 1.66x10-19 * =(6.626x10-24 v=0.084x10 m =0.084x10 cm Therefore, the energy separation in centimeter inverse is 0.084x10 cm