Practice problems for these concepts can be found at:

- Spectroscopy, Light, and Electrons Multiple Choice Review Questions for AP Chemistry
- Spectroscopy, Light, and Electrons Free-Response Questions for AP Chemistry

Bohr's model worked well for hydrogen, the simplest atom, but didn't work very well for any others. In the early 1900s Schröinger developed a more involved model and set of equations that better described atoms by using quantum mechanical concepts. His model introduced a mathematical description of the electron's motion called a **wave function** or **atomic orbital**. Squaring the wave function (orbital) gives the volume of space in which the probability of finding the electron is high. This is commonly referred to as the **electron cloud (electron density)**.

Schröinger's equation required the use of three **quantum numbers** to describe each electron within an atom, corresponding to the orbital size, shape, and orientation in space. It was also found that a quantum number concerning the spin of the electron was needed.

The first quantum number is the **principal quantum number ( n)**. It describes the energy (related to size) of the orbital and relative distance from the nucleus. The allowed (by the mathematics of the Schröinger equation) values are positive integers (1, 2, 3, 4, etc.). The smaller the value of n, the closer the orbital is to the nucleus. The number n is sometimes called the atom's

**shell**.

The second quantum number is the **angular momentum quantum number ( l)**. Its value is related to the principal quantum number and has allowed values of 0 up to (

*n*– 1). For example, if

*n*= 3, then the possible values of

*l*would be 0,1, and 2 (3 – 1). This value of

*l*defines the shape of the orbital:

- If
*l*= 0, the orbital is called an s orbital and has a spherical shape with the nucleus at the center of the sphere. The greater the value of n, the larger the sphere. - If
*l*= 1, the orbital is called a p orbital and has two lobes of high electron density on either side of the nucleus. This makes for an hourglass or dumbbell shape. - If
*l*= 2, the orbital is a d orbital and can have variety of shapes. - If
*l*= 3, the orbital is an f orbital, with more complex shapes.

Figure 10.3 shows the shapes of the s, p, and d orbitals. These are sometimes called **sublevels** or **subshells**.

The third quantum number is the **magnetic quantum number ( m_{l})**. It describes the orientation of the orbital around the nucleus. The possible values of

*m*

_{1}depend on the value of the angular momentum quantum number,

*l*. The allowed values for m

_{l}are –

*l*through zero to +

*l*. For example, for

*l*= 2 the possible values of ml would be –2, –1, 0, +1, +2. This is why, for example, if

*l*= 1 (a p orbital), then there are three p orbitals corresponding to ml values of –1, 0, +1. This is also shown in Figure 10.3.

The fourth quantum number, the **spin quantum number ( m_{s})**, indicates the direction the electron is spinning. There are only two possible values for

*m*, +

_{s}^{1}/

_{2}and –

^{1}/

_{2}.

The quantum numbers for the six electrons in carbon would be:

Therefore, the electron configuration of carbon is 1s^{2}2s^{2}2p^{2}.

Practice problems for these concepts can be found at:

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