Question

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

Short answer

For the 4s electrons in titanium, the possible sets of quantum numbers are: 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 the 3d electrons in titanium, the possible sets of quantum numbers are: 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. \(n = 3\), \(l = 2\), \(m_l = 2\), \(m_s = -1/2\) The two 3d electrons can occupy any of the above combinations of quantum numbers.

Step-by-step solution

Review the Four Quantum Numbers

There are four quantum numbers that describe an electron in an atom: 1. Principal quantum number (n): It determines the electron's energy and distance from the nucleus; it can have integer values (n = 1, 2, 3, ...). 2. Azimuthal quantum number (l): It defines the electron's orbital shape and depends on the principal quantum number; it can have integer values from 0 to n-1. 3. Magnetic quantum number (m_l): It describes the orientation of the orbital in space; it can have integer values from -l to +l. 4. Spin quantum number (m_s): It represents the electron's spin, which can be either +1/2 (up) or -1/2 (down).

Determine the quantum numbers for 4s electrons

For 4s electrons, n = 4 (since it is the 4th shell), and s implies that l = 0 (since s, p, d, and f correspond to l = 0, 1, 2, and 3, respectively). As the azimuthal quantum number l = 0, the magnetic quantum number m_l must also be 0, as it ranges from -l to +l. Each 4s orbital can hold a maximum of two electrons with opposite spins (+1/2 and -1/2), and the titanium atom has only two 4s electrons. Therefore, for 4s electrons, the possible sets of quantum numbers in titanium are: 1. n = 4, l = 0, m_l = 0, m_s = +1/2 2. n = 4, l = 0, m_l = 0, m_s = -1/2

Determine the quantum numbers for 3d electrons

For 3d electrons, n = 3 (since it is the 3rd shell), and d implies that l = 2 (since s, p, d, and f correspond to l = 0, 1, 2, and 3, respectively). With the azimuthal quantum number l = 2, the magnetic quantum number m_l can have the following values: -2, -1, 0, 1, and 2. Each 3d orbital can also hold a maximum of two electrons with opposite spins (+1/2 and -1/2), and the titanium atom has only two 3d electrons. Therefore, for 3d electrons, the possible sets of quantum numbers in titanium are: 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 The two 3d electrons can occupy any of the above combinations of quantum numbers. In summary, for the 4s and 3d electrons in titanium, we have found the possible sets of values for the four quantum numbers.