Problem 98 - Chemistry An Atoms First Approach

Question

Give a possible set of values of the four quantum numbers for the \(4 s\) and \(3 d\) electrons in titanium.

Step-by-Step Solution

Answer

For the 4s electrons in titanium, the possible sets of quantum numbers are (n = 4, l = 0, m_l = 0, m_s = +1/2) and (n = 4, l = 0, m_l = 0, m_s = -1/2). For the 3d electrons, the possible sets of quantum numbers are: n = 3, l = 2, and m_l ranges from -2 to 2 with m_s being either +1/2 or -1/2, yielding ten possible sets of values in total.

Step 1: Identify the electronic configuration

To determine the set of quantum numbers for the 4s and 3d electrons, we first need to find the electronic configuration of titanium. Titanium (Ti) has an atomic number of 22. Therefore, it has 22 electrons. The electronic configuration for titanium is: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d² Now, we'll focus on the 4s and 3d electrons.

Step 2: Determine the principal quantum number (n)

The principal quantum number (n) is directly related to the energy level of the electron. In this case, the 4s electrons have n = 4, while the 3d electrons have n = 3.

Step 3: Determine the azimuthal quantum number (l)

The azimuthal quantum number (l) is related to the shape of the orbital. The values for l depend on the value of n and range from 0 to (n-1). Here, we already know the orbitals: s and d. The s orbital corresponds to l = 0, and the d orbital corresponds to l = 2. So for 4s electrons, l = 0, and for 3d electrons, l = 2.

Step 4: Determine the magnetic quantum number (m_l)

The magnetic quantum number (m_l) describes the orientation of the orbital. The values for m_l depend on the value of l and range from -l to +l, including 0. For 4s electrons (l = 0), there is only one possible value for m_l: 0. For 3d electrons (l = 2), there are five possible values for m_l: -2, -1, 0, 1, and 2.

Step 5: Determine the spin quantum number (m_s)

The spin quantum number (m_s) describes the intrinsic angular momentum of the electron. There are two possible values for m_s: +1/2 and -1/2, representing spin-up and spin-down electrons, respectively.

Step 6: Find the possible sets of quantum numbers

Now, we can combine the information to find the possible sets of quantum numbers for the 4s and 3d electrons in titanium. For 4s electrons: 1. n = 4, l = 0, m_l = 0, m_s = +1/2 2. n = 4, l = 0, m_l = 0, m_s = -1/2 For 3d electrons: 1. n = 3, l = 2, m_l = -2, m_s = +1/2 2. n = 3, l = 2, m_l = -2, m_s = -1/2 3. n = 3, l = 2, m_l = -1, m_s = +1/2 4. n = 3, l = 2, m_l = -1, m_s = -1/2 5. n = 3, l = 2, m_l = 0, m_s = +1/2 6. n = 3, l = 2, m_l = 0, m_s = -1/2 7. n = 3, l = 2, m_l = 1, m_s = +1/2 8. n = 3, l = 2, m_l = 1, m_s = -1/2 9. n = 3, l = 2, m_l = 2, m_s = +1/2 10. n = 3, l = 2, m_l = 2, m_s = -1/2 These are the possible sets of values for the four quantum numbers for the 4s and 3d electrons in titanium.