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Solutions 16

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    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?


    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


    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. \[P = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{a_0}^{\infty} \dfrac{3}{96\pi}Cos^2(\theta )Sin(\theta )\rho^4 exp(-\rho) d\rho d\theta d\phi \]


    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)


    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.


    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


    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


    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


    3 13 5 18
    2 3 1 4
    1 0 0 0


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


    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.


    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.

    Solutions 16 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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