If a H_2 molecule rotates in the plane of a crystalline surface, it can be appro
ID: 1061538 • Letter: I
Question
If a H_2 molecule rotates in the plane of a crystalline surface, it can be approximated as a two-dimensional rigid rotor. Calculate in kJ/mol the lowest energy rotational transition for such system. The spacing between lines in the pure rotational spectra of^7Li^1H is 4.05 times 10^11 s^-1. Calculate the bond length of this molecules. Calculate the mean values of the radius, , at which you would find the electron in the H atom if the electron is in the state n = 1, l = 0, and m = 0. Is psi(1, 2) = 1s(1) alpha(1) 1s(2) alpha(2) + 1s(2) + 1s(2) alpha(2) 1s(1) alpha(1) an eigenfunction of the S? If so, what is its eigenvalues M_s? In 1s^- 2s^1 excited state of helium, the Coulomb integral is equal to 0.420 Eh and the exchange integral is equal to 0.044 Eh. Estimate the average distance in angstrom between the two electrons in the singlet state, in the triplet state, using these numbers and assuming that = ^-1. Apply the variational trial function x^2(a-x) for x between 0 and a to the particle in the box and estimate the ground-state energy. Compare this energy to the exact energy for the ground state of the particle in the box.Explanation / Answer
Ans-2.
Spacing between lines is (2B) = 4.05*10^11
Where B is the rotational Constant.
Reduced mass= M1M2/(M1+M2)
In this case M1=Li
M2= H
So Reduced mass()= 7/8
According to the formula-
2B= (h2/4²) /R²
Where h is plank's constant having value 6.6×10-34
and R= Bond Length
After putting values and calculate- R=3.13 e-39
Ans-3. Hydrogen wave function at n=1, l=0, m=0 is
100 =(1÷a03 )1/2 e-p
Where p=r/a0
Now probability density of this function can be founf by squaring it and multiply with a volume element
i.e. 100 2 = [(1÷a03 )1/2 e-p ]2 . 4r2dr= 4/ao3 r2 e-2r/ao dr
Nor mean value r
r= r(dP/dr).dr= 4/ao3 r3 e-2r/ao dr
0 0
After solving by integration by Parts-
we get r= 3ao/2
Ans- 6. Hamiltonian of partical in a box= H=[(-h2/2m).d2/dx2 ]
Variation Method
Energy= = Hd÷d
In the question = x2
and limit is from 0 to a
By putting values energy can be calculate.