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# Solutions 16

## Q16.1

Which quantum numbers influence the energy of a specific electron in a hydrogen atom? What about for a lithium atom? If there is a different, why?

## S16.1

1. Only the n quantum number determines the energy of an electron in a hydrogen atom.
2. In lithium it is again only the n quantum number that determines the electron energy since all three electrons are in a s-orbital

## Q16.2

Write the integrals/equations that need to be solved (do not solve) to calculate the following values:

1. Probability of finding an electron in hydrogen in the 1s orbital at a radius between $$a_o$$ and $$2a_o$$
2. Probability of finding an electron in hydrogen in the 2s orbital within a radius of $$\dfrac{a_o}{2}$$
3. The most probable radius for an electron in hydrogen in the 2s orbital.
4. The probability of finding an electron in the $$n=1$$ and $$l=0$$ of being outside the Bohr Radius.
5. The probability of finding an electron in the $$n=2$$ and $$l=1$$ of being outside the Bohr Radius.

When integrating using spherical coordinates you need to multiply the integral by $$r^2Sin(\theta)$$. With integrals that just involve the $$r$$ coordinate that can be replaced by $$4\pi r^2$$.

1. $P = \int_{a_0}^{2a_0}2\rho^2exp(-2\rho)d\rho$
2. $P = \int_{0}^{\dfrac{a_0}{2}}\dfrac{1}{2}\rho^2(1 - \dfrac{\rho}{2})^2exp(-2\rho)d\rho$
3. $\dfrac{d\psi^2}{d\rho} = 0$
4. $\dfrac{d\psi^2}{d\rho} = \dfrac{d\dfrac{1}{2}\rho (1 - \dfrac{\rho}{2})^2exp(-2\rho)}{d\rho} = 0$
5. $P = \int_{a_0}^{\infty}2\rho^2exp(-2\rho)d\rho$
6. $P = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{a_0}^{\infty} \dfrac{3}{96\pi}Cos^2(\theta )Sin(\theta )\rho^4 exp(-\rho) d\rho d\theta d\phi$

## Q16.3

How many electrons could be held in the second shell of an atom if the spin quantum number $$m_s$$ could have three values instead of just two? (Hint: Consider the Pauli exclusion principle)

## S16.3

The second shell has both the s- and p-subshells with one and three orbitals, repsectively. If each of the four orbitals could hold three electrons, then a total of 12 electrons could be added to the second shell, i.e. 4 orbitals times 3 electrons = 12 electrons.

## Q16.4

Write a set of quantum numbers for each of the electrons with an n of 4 in a Se atom.

 n l $$m_l$$ $$m_s$$ 4 0 0 1/2 4 0 0 -1/2 4 1 0 1/2 4 1 0 -1/2 4 1 -1 +/-1/2 4 1 1 +/-1/2

## Q16.5

Use an orbital diagram to describe the electron configuration of the valence shell of each of the following atoms:

1. N
2. Si
3. Fe
4. Te
5. Mo

## Q16.6

What are number of radial, angular and total nodes for all wavefunctions in the first three shells ($$n \le 3$$) of hydrogen atom. What are the general equations relating the number of radial angular and total nodes as a function of quantum numbers.

 n Angular Radial Total 3 13 5 18 2 3 1 4 1 0 0 0

## S16.6

The total number of nodes per orbital is n-1 the number of angular nodes is l.

## Q16.7

Explain the general trend and rational between $$Z$$ and $$Z_{eff}$$ for the outermost (valence) electron and core electron (i.e., $$n=1$$) in the first 20 elements of the periodic table.

## S16.7

For the core electrons Z is approximately equal to $$Z_{eff}$$, for the outermost electrons Z is larger than $$Z_{eff}$$, but $$Z_{eff}$$ increases as you move farther right across the rows of the periodic table.