Question

In an atom, what is the maximum number of electrons that can have the following quantum numbers? n=3, I=2, mI=-2 and n=4, I=1, mI=+1 Please explain and thank you so much!

Answer 1

Quantum Numbers

The Bohr model was a one-dimensional model that used one quantum number to describe the distribution of electrons in the atom. The only information that was important was the size of the orbit, which was described by the n quantum number. Schr�dinger's model allowed the electron to occupy three-dimensional space. It therefore required three coordinates, or three quantum numbers, to describe the orbitals in which electrons can be found.

The three coordinates that come from Schr�dinger's wave equations are the principal (n), angular (l), and magnetic (m) quantum numbers. These quantum numbers describe the size, shape, and orientation in space of the orbitals on an atom.

The principal quantum number (n) describes the size of the orbital. Orbitals for which n = 2 are larger than those for which n = 1, for example. Because they have opposite electrical charges, electrons are attracted to the nucleus of the atom. Energy must therefore be absorbed to excite an electron from an orbital in which the electron is close to the nucleus (n = 1) into an orbital in which it is further from the nucleus (n = 2). The principal quantum number therefore indirectly describes the energy of an orbital.

The angular quantum number (l) describes the shape of the orbital. Orbitals have shapes that are best described as spherical (l = 0), polar (l = 1), or cloverleaf (l = 2). They can even take on more complex shapes as the value of the angular quantum number becomes larger.

There is only one way in which a sphere (l = 0) can be oriented in space. Orbitals that have polar (l = 1) or cloverleaf (l = 2) shapes, however, can point in different directions. We therefore need a third quantum number, known as the magnetic quantum number (m), to describe the orientation in space of a particular orbital. (It is called the magnetic quantum number because the effect of different orientations of orbitals was first observed in the presence of a magnetic field.)

Shells and Subshells of Orbitals

Orbitals that have the same value of the principal quantum number form a shell. Orbitals within a shell are divided into subshells that have the same value of the angular quantum number. Chemists describe the shell and subshell in which an orbital belongs with a two-character code such as 2p or 4f. The first character indicates the shell (n = 2 or n = 4). The second character identifies the subshell. By convention, the following lowercase letters are used to indicate different subshells.

s:		l = 0
p:		l = 1
d:		l = 2
f:		l = 3

Although there is no pattern in the first four letters (s, p, d, f), the letters progress alphabetically from that point (g, h, and so on). Some of the allowed combinations of the n and l quantum numbers are shown in the figure below.

The third rule limiting allowed combinations of the n, l, and m quantum numbers has an important consequence. It forces the number of subshells in a shell to be equal to the principal quantum number for the shell. The n = 3 shell, for example, contains three subshells: the 3s, 3p, and 3d orbitals.

Possible Combinations of Quantum Numbers

There is only one orbital in the n = 1 shell because there is only one way in which a sphere can be oriented in space. The only allowed combination of quantum numbers for which n = 1 is the following.

n		l		m
1		0		0		1s

There are four orbitals in the n = 2 shell.

n		l		m
2		0		0		2s

2		1		-1
2		1		0		2p
2		1		1

There is only one orbital in the 2s subshell. But, there are three orbitals in the 2p subshell because there are three directions in which a p orbital can point. One of these orbitals is oriented along the X axis, another along the Y axis, and the third along the Z axis of a coordinate system, as shown in the figure below. These orbitals are therefore known as the 2p_x, 2p_y, and 2p_z orbitals.

There are nine orbitals in the n = 3 shell.

n		l		m
3		0		0		3s
3		1		-1
3		1		0		3p
3		1		1
3		2		-2
3		2		-1		3d
3		2		0
3		2		1
3		2		2

There is one orbital in the 3s subshell and three orbitals in the 3p subshell. The n = 3 shell, however, also includes 3d orbitals.

The five different orientations of orbitals in the 3d subshell are shown in the figure below. One of these orbitals lies in the XY plane of an XYZ coordinate system and is called the 3d_xy orbital. The 3d_xz and 3d_yz orbitals have the same shape, but they lie between the axes of the coordinate system in the XZ and YZ planes. The fourth orbital in this subshell lies along the X and Y axes and is called the 3d_x²_-y² orbital. Most of the space occupied by the fifth orbital lies along the Z axis and this orbital is called the 3d_z² orbital.

The number of orbitals in a shell is the square of the principal quantum number: 1² = 1, 2² = 4, 3² = 9. There is one orbital in an s subshell (l = 0), three orbitals in a p subshell (l = 1), and five orbitals in a dsubshell (l = 2). The number of orbitals in a subshell is therefore 2(l) + 1.

Before we can use these orbitals we need to know the number of electrons that can occupy an orbital and how they can be distinguished from one another. Experimental evidence suggests that an orbital can hold no more than two electrons.

To distinguish between the two electrons in an orbital, we need a fourth quantum number. This is called the spin quantum number (s) because electrons behave as if they were spinning in either a clockwise or counterclockwise fashion. One of the electrons in an orbital is arbitrarily assigned an s quantum number of +1/2, the other is assigned an s quantum number of -1/2. Thus, it takes three quantum numbers to define an orbital but four quantum numbers to identify one of the electrons that can occupy the orbital.

The allowed combinations of n, l, and m quantum numbers for the first four shells are given in the table below. For each of these orbitals, there are two allowed values of the spin quantum number, s.