Homework #6

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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.

What is the maximum number of electrons per orbital?
Each electron in the \(7s\) orbital would have what four quantum numbers?
If \(n=4\), what are the values of the second quantum number?
The five \(3d\) orbitals can collectively hold how many electrons?
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.

Sn
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)?

\(n = 4, l = 2, m_l = 1, m_s = 1/2\)
\(n = 4, l = 1, m_l = 1, m_s = 1/2\)
\(n = 3, l = 1, m_l = 1, m_s = 1/2\)
\(n = 4, l = 3, m_l = 1, m_s = 1/2\)
\(n = 4, l = 1, m_l = 1/2, m_s = 0\)

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 \(\sigma\).)