openstax physics answer hydrogen h2 molecules are kept at 300.0 k in a cubical container with a side length of 20.0 cm. assume that you can treat the molecules as though they were moving in a one-dimensional box.

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16-12-2022
Chemistry

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openstax physics answer hydrogen h2 molecules are kept at 300.0 k in a cubical container with a side length of 20.0 cm. assume that you can treat the molecules as though they were moving in a one-dimensional box.

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abhinavjh321 abhinavjh321 · Answer 1 · 2022-12-20T21:46:05-05:00

The ground state energy of the hydrogen molecule in the container is 2.56x10-24eV at 300K and the corresponding quantum number n of the quantum state that would correspond to this thermal energy is 2.2x10-8J

Given temperature of H2 molecules(T) =300K

length of cubic container (l) = 20cm

(a) A quantum particle in a box have discrete values of energy.

En = n^2π^2h^2/2mL^2 where L is the width of container n is energy level m is the mass of particle and h is plancks constant.

Mass of H2 = 3.35x10-27kg as 1 Na molecule = 2x1.008gml and n = 1

h = 1.05x10-34Js

So En = 1^2xπ^2x(1.05x10-34)^2/2x3.35x10-27x(0.2)^2 = 2.56x10-24eV

(b) We know that E = KbT/2 then the thermal energy E = 1.380x10-23x300/2 = 207x10-23J

From En = n^2π^2h^2/2mL^2 = we have the level of H2 molecule as:

n= √Enx2xmxL^2/π^2xh^2 = L/πh√Enx2xm = (0.2)√207x10-23x2x3.35x10-27/3.14x1.05x10-34 = 7.44x10-25/3.297x10-17 = 2.2x10-8

Hence the energy level corresponds to this quantum state (n) = 2.2x10-8J

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complete question: Hydrogen molecules H2 are kept at in acubical container with a side length of 20cm. Assume that you can treat the molecules as though they were moving in a one-dimensional box. (a) Find the ground state energy of the hydrogen molecule in the container. (b) Assume that the molecule has a thermal energy given by and find the corresponding quantum number of the quantum state that would correspond to this thermal energy.