# Homework #6


Show your work for calculations.

## Q1

Draw $$2s$$ orbital with a radial node specified by a dashed line.

## Q2

Draw $$2p_z$$ orbital with the x, y, and z axes.

## Q3

Draw $$3d_{xy}$$, $$3d_{x^2-y^2}$$, and $$3d_{z^2}$$ orbitals with the x, y, and z axes.

## Q4

Answer the questions below.

1. What is the maximum number of electrons per orbital?
2. Each electron in the $$7s$$ orbital would have what four quantum numbers?
3. If $$n=4$$, what are the values of the second quantum number?
4. The five $$3d$$ orbitals can collectively hold how many electrons?
5. How many total electrons are in all the $$p$$ orbitals in an atom of Sulfur (S)?

## Q5

Depict the ground state electron configuration of the following atoms by using the orbital diagram with noble-gas-core abbreviation.

1. Sn
2. Mo (Hint: A half-filled $$d$$ subshell is particularly stable.)

## Q6

Depict the ground state electron configuration of titanium (Ti) by using condensed $$spdf$$ notation without noble-gas-core abbreviation.

## Q7

Which of the following could be the four quantum numbers of the last electron of arsenic (As)?

1. $$n = 4, l = 2, m_l = 1, m_s = 1/2$$
2. $$n = 4, l = 1, m_l = 1, m_s = 1/2$$
3. $$n = 3, l = 1, m_l = 1, m_s = 1/2$$
4. $$n = 4, l = 3, m_l = 1, m_s = 1/2$$
5. $$n = 4, l = 1, m_l = 1/2, m_s = 0$$

## Q8

At what distance from the nucleus does the radial node of $$2s$$ orbital of $$Li^{2+}$$ occur? Express the answer in terms of $$a_0$$(Bohr radius). Use the following equations describing the wave function ($$\psi_{2s}$$).

$\psi_{2s}=R_{2s}Y_{s}$

$R_{2s}=\dfrac{1}{2\sqrt{2}}\Big(\dfrac{Z}{a_0}\Big)^{3/2}(2-\sigma)e^{-\sigma/2},\hspace{1cm}\sigma=\dfrac{2Z}{na_0}r$

$Y_{s}=\sqrt{\dfrac{1}{4\pi}}$

(Hint)

• $$R_{2s}$$ is a function of distance $$r$$, but is expressed in terms of $$\sigma$$. $$\sigma$$ is a constant ($$\dfrac{2Z}{na_0}$$) times $$r$$, where $$Z$$, $$n$$, and $$a_0$$ represent the atomic number, the principal quantum number, and the Bohr radius($$=5.29\times10^{-11} m$$), respectively.
• The radial node occurs at the distance where the function $$R_{2s}$$ is equal to zero. Note that $$e^{-\sigma/2}>0$$ for all $$\sigma$$.)

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