Question

For each electron in a ground-state Beryllium atom, select the set of quantum numbers that represents it. Check all that apply.

n = 1, l= 0, ml= –1, ms= 1/2
n = 2, l= 0, ml= 0, ms= –1
n = 1, l= 0, ml= 0, ms= 1/2
n = 1, l= 0, ml= 0, ms= –1/2
n = 2, l= 0, ml= 0, ms= 1/2
n = 2, l= 1, ml= 0, ms= –1/2
n = 2, l= 1, ml= 1, ms= 1/2
n = 2, l= 0, ml= 0, ms= –1/2

Answer 1

Concepts and reason

The concept used to solve the given problem is the application of the theory of quantum numbers. There a total of four quantum numbers, based on which the electron movement trajectory can be defined. The quantum numbers for an electron are unique and different for any two electrons residing in an orbital.

Fundamentals

The four quantum numbers are orbital angular momentum quantum number \((l)\), principal quantum number \((n)\),

electron spin quantum number \(\left(m_{s}\right)\) and magnetic quantum number \(\left(m_{l}\right)\). Orbital angular momentum quantum number is helpful in the determination of the shape of an orbital. The magnetic spin quantum number helps in the determination of the orientation of the electron in the subshell. Electron spin quantum number defines the electronic spin of the electron in the orbital.

 

The electron configuration of beryllium (Be), having atomic number 4, is mentioned as follows:

\(\mathrm{Be}=1 s^{2} 2 s^{2}\)

The electronic configuration is the electron distribution of a particular atom in its molecular orbitals. The occupancy order for the orbitals has been shown as follows:

\(1 s<2 s<2 p<3 s<3 p<4 s<3 d<4 p<5 s<4 d<5 p<6 s<4 f<5 d<6 p<7 s<5 f<6 d<7 p\)

Furthermore, the electrons are filled in increasing order of these orbitals. There are three basic rules for electron configuration: Hund's rule, Aufbau's principle, and the Pauli-exclusion principle.

 

The principal quantum number is needed to be determined first, followed by the orbital angular momentum quantum number and magnetic quantum number. Thus, the quantum numbers for \(1 s^{1}\) electron can be determined as follows:

Principal quantum number \((n)=1\) Orbitalangularmomentumquantumnumber \((l)=n--1\) Substitute the value of \(n\) as 1 in the above given equation and the value of \(l\) can be calculated as follows:

Orbitalangularmomentumquantumnumber \((l)=n--1\)

$$ \begin{aligned} =& 1-1 \\ =& 0 \end{aligned} $$

Magneticquantumnumber \(\left(m_{l}\right)=0\) Electronicspinquantumnumber \(\left(m_{s}\right)=+1 / 2\)

The magnetic quantum number is related to the orbital angular quantum number, which depends on the principal quantum number. Furthermore, the electron spin quantum number depends not on any of the first three mentioned quantum numbers. The shell number gives the value of the principal quantum number. Thus, for electron residing in \(1 s^{1}\) have the value of \(n\) ' equal to 1 . The value of the orbital angular quantum number is given by subtracting 1 from the value of the principal quantum number. The range of \(-l\) to \(+l\) gives the value of a magnetic quantum number. The value of electron spin quantum number is either \(+1 / 2\) or \(-1 / 2\). However, in an orbital, the values are of the opposite nature.

 

Quantum numbers for \(1 s^{2}\) electron can be written as follows:

Principalquantumnumber \((n)=1\) Orbitalangularmomentumquantumnumber \((l)=n--1\) Substitute the value of \(n\) as 1 in the above given equation and the value of \(l\) can be calculated as follows:

Orbitalangularmomentumquantumnumber \((l)=n--1\)

$$ \begin{array}{r} =1-1 \\ =0 \end{array} $$

Magneticquantumnumber \(\left(m_{l}\right)=0\) Electronicspinquantumnumber \(\left(m_{s}\right)=-1 / 2\)

The shell number gives the value of the principal quantum number. Thus, for electron residing in \(1 s^{1}\) have the value of \(^{\prime} n\) ' equal to 1 . The value of the orbital angular quantum number is given by subtracting 1 from the value of the principal quantum number. The magnetic quantum number's value is given by the range of \(-l\) to \(+l\). The value of electron spin quantum number is either \(+1 / 2\) or \(-1 / 2\). However, in an orbital, the values are of opposite nature. Further, the value of \(m_{s}\) is \(-1 / 2\) as \(1 s^{1}\) electron has been designated as \(+1 / 2\)

 

Quantum numbers for \(2 s^{1}\) electron can be written as follows:

Principalquantumnumber \((n)=2\) Orbitalangularmomentumquantumnumber \((l)=n--1\) Substitute the value of \(n\) as 2 in the above given equation and the value of \(l\) can be calculated as follows:

Orbitalangularmomentumquantumnumber \((l)=n--1\)

$$ \begin{array}{r} =2-1 \\ =1 \end{array} $$

Magneticquantumnumber \(\left(m_{l}\right)=3\) Electronicspinquantumnumber \(\left(m_{s}\right)=+1 / 2\)

The shell number gives the principal quantum number's value. Thus, for electron residing in \(2 s^{1}\) have the value of 'n' equal to 2 . The value of the orbital angular quantum number is given by subtracting 1 from the principal quantum number's value. The magnetic quantum number's value is given by the range of -1 to \(+1 .\) The value of electron spin quantum number is either \(+1 / 2\) or \(-1 / 2 .\) However, in an orbital, the values are of opposite nature.

 

Quantum numbers for \(2 \mathrm{~s} 2\) electron can be written as follows:

Principalquantumnumber \((n)=2\) Orbitalangularmomentumquantumnumber \((l)=n--1\) Substitute the value of \(n\) as 2 in the above given equation and the value of \(l\) can be calculated as follows:

Orbitalangularmomentumquantumnumber \((l)=n--1\)

$$ \begin{aligned} =& 2-1 \\ &=1 \end{aligned} $$

Magneticquantumnumber \(\left(m_{l}\right)=3\) Electronicspinquantumnumber \(\left(m_{s}\right)=-1 / 2\)

The quantum numbers for different electron in beryllium have been mentioned as follows:

\(1 s^{1}\) electron: \(n=1 ; l=0 ; m_{l}=0\) and \(m_{s}=+1 / 2\)

\(1 s^{2}\) electron: \(n=1 ; l=0 ; m_{l}=0\) and \(m_{s}=-1 / 2\)

\(2 s^{1}\) electron: \(n=2 ; l=0 ; m_{l}=0\) and \(m_{s}=+1 / 2\)

\(2 s^{2}\) electron: \(n=2 ; l=0 ; m_{l}=0\) and \(m_{s}=-1 / 2\)

The shell number gives the principal quantum number's value. Thus, for electron residing in \(1 s^{1}\) have the value of \(n\) ' equal to 1 . The value of the orbital angular quantum number is given by subtracting 1 from the value of the principal quantum number. The magnetic quantum number's value is given by the range of \(-l\) to \(+l\). The value of electron spin quantum number is either \(+1 / 2\) or \(-1 / 2 .\) However, in an orbital, the values are of opposite nature. Further, the value of \(m_{s}\) is \(-1 / 2\) as \(1 s^{1}\) electron has been designated as \(+1 / 2\).