Medium
How many electrons in an atom may have the following quantum numbers?
$(a)$ $n=4, m_{s}=-\frac{1}{2}$
$(b)$ $n=3, l=0$
$(a)$ $n=4, m_{s}=-\frac{1}{2}$
$(b)$ $n=3, l=0$
(A) The total number of electrons in a shell with principal quantum number $n$ is $2n^{2}$.
For $n=4$,the total number of electrons is $2(4)^{2} = 32$.
In a neutral atom with a filled shell,half of the electrons have spin $m_{s}=+\frac{1}{2}$ and half have spin $m_{s}=-\frac{1}{2}$.
Therefore,the number of electrons with $n=4$ and $m_{s}=-\frac{1}{2}$ is $\frac{32}{2} = 16$.
$(b)$ The quantum numbers $n=3$ and $l=0$ correspond to the $3s$ orbital.
Since an $s$ orbital can hold a maximum of $2$ electrons,the number of electrons with $n=3$ and $l=0$ is $2$.
For $n=4$,the total number of electrons is $2(4)^{2} = 32$.
In a neutral atom with a filled shell,half of the electrons have spin $m_{s}=+\frac{1}{2}$ and half have spin $m_{s}=-\frac{1}{2}$.
Therefore,the number of electrons with $n=4$ and $m_{s}=-\frac{1}{2}$ is $\frac{32}{2} = 16$.
$(b)$ The quantum numbers $n=3$ and $l=0$ correspond to the $3s$ orbital.
Since an $s$ orbital can hold a maximum of $2$ electrons,the number of electrons with $n=3$ and $l=0$ is $2$.
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The hydrogen-like species $Li^{2+}$ is in a spherically symmetric state $S_1$ with one radial node. Upon absorbing light,the ion undergoes transition to a state $S_2$. The state $S_2$ has one radial node and its energy is equal to the ground state energy of the hydrogen atom.
$1.$ The state $S_1$ is
$(A)$ $1s$ $(B)$ $2s$ $(C)$ $2p$ $(D)$ $3s$
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$(A)$ $0.75$ $(B)$ $1.50$ $(C)$ $2.25$ $(D)$ $4.50$
$3.$ The orbital angular momentum quantum number of the state $S_2$ is
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$(A)$ $0.75$ $(B)$ $1.50$ $(C)$ $2.25$ $(D)$ $4.50$
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$(A)$ $0$ $(B)$ $1$ $(C)$ $2$ $(D)$ $3$
Give the answer for questions $1, 2$ and $3.$
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View SolutionArrange the electrons represented by the following sets of quantum numbers in the decreasing order of energy.
$(i) \ n = 4, l = 0, m_l = 0, m_s = +\frac{1}{2}$
$(ii) \ n = 3, l = 1, m_l = 1, m_s = -\frac{1}{2}$
$(iii) \ n = 3, l = 2, m_l = 0, m_s = +\frac{1}{2}$
$(iv) \ n = 3, l = 0, m_l = 0, m_s = -\frac{1}{2}$
$(i) \ n = 4, l = 0, m_l = 0, m_s = +\frac{1}{2}$
$(ii) \ n = 3, l = 1, m_l = 1, m_s = -\frac{1}{2}$
$(iii) \ n = 3, l = 2, m_l = 0, m_s = +\frac{1}{2}$
$(iv) \ n = 3, l = 0, m_l = 0, m_s = -\frac{1}{2}$
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