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Chapter 7: Problem 26

The periodic table consists of four blocks of elements that correspond to \(s, p, d\), and \(f\) orbitals being filled. After \(f\) orbitals come \(g\) and \(h\) orbitals. In theory, if a \(g\) block and an \(h\) block of elements existed, how long would the rows of \(g\) and \(h\) elements be in this theoretical periodic table?

Short Answer

Expert verified
In a theoretical periodic table, the rows of \(g\) and \(h\) elements would be 18 and 22 elements long, respectively. This is based on the number of orbitals for each orbital type (\(g\) orbitals have 9 orbitals, and \(h\) orbitals have 11 orbitals) and considering that each orbital can accommodate two electrons due to their spin.

Step by step solution

01

Determine the maximum azimuthal quantum number (l) for the \(g\) and \(h\) orbitals

We already know the azimuthal quantum numbers (l) for some blocks of orbitals: - For the \(s\) orbitals, \(l = 0\) - For the \(p\) orbitals, \(l = 1\) - For the \(d\) orbitals, \(l = 2\) - For the \(f\) orbitals, \(l = 3\) Following the same pattern, we can assign the azimuthal quantum numbers for the \(g\) and \(h\) orbitals: - For the \(g\) orbitals, \(l = 4\) - For the \(h\) orbitals, \(l = 5\)
02

Determine the maximum magnetic quantum number (m_l) for each orbital type

For each azimuthal quantum number (l), the range of magnetic quantum numbers (m_l) goes from \(-l\) to \(+l\), both inclusive. This range will help us calculate the number of orbitals for each orbital type. For example, for the \(s\) orbitals with \(l = 0\), the magnetic quantum number (m_l) is 0, so there is only one orbital. - For the \(g\) orbitals, the magnetic quantum number (m_l) ranges from -4 to 4: \(m_l \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}\) - For the \(h\) orbitals, the magnetic quantum number (m_l) ranges from -5 to 5: \(m_l \in \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}\)
03

Calculate the number of orbitals for each orbital type

Count the number of magnetic quantum numbers (m_l) in each range to find the number of orbitals: - For the \(g\) orbitals, there are 9 orbitals. - For the \(h\) orbitals, there are 11 orbitals.
04

Determine the length of rows for \(g\) and \(h\) elements

Since there are two elements for each orbital due to electron spin (up and down), we find the number of elements in each block: - For the \(g\) orbitals, there would be \(9 \times 2 = 18\) elements in a row. - For the \(h\) orbitals, there would be \(11 \times 2 = 22\) elements in a row. So, in this theoretical periodic table, the rows of \(g\) and \(h\) elements would be 18 and 22 elements long, respectively.

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