In a hydrogen atom, the electron is in the \(n=5\) state. Which of the following sets could correspond to the \(\ell, m\) states of the electron? a) 5,-3 b) 4,-5 c) 3,-2 d) 4,-6

For a given principal quantum number \(n\), the possible values of the angular momentum quantum number \(\ell\) are \(0,1,2, \dots,n-1\). And for a given \(\ell\) value, the possible values of the magnetic quantum number \(m\) are \(-\ell, -(\ell-1), \dots, 0, \dots, (\ell-1), \ell\).

We will now compare each option with the quantum number rules mentioned in Step 1. a) For option (a), we have \(\ell = 5\) and \(m = -3\). Since the possible values for \(\ell\) are \(0, 1, 2, 3, 4\) when \(n=5\), this option is invalid as \(\ell\) is out of range. b) For option (b), we have \(\ell = 4\) and \(m = -5\). Here, \(\ell = 4\) is within the range for \(n = 5\). However, the possible values of \(m\) for \(\ell = 4\) are \(-4, -3, -2, -1, 0, 1, 2, 3, 4\). As \(m=-5\) is not within this range, this option is also invalid. c) For option (c), we have \(\ell = 3\) and \(m = -2\). The \(\ell\) value is within the range for \(n = 5\). The possible values of \(m\) for \(\ell = 3\) are \(-3, -2, -1, 0, 1, 2, 3\). As \(m=-2\) is within this range, this option is valid. d) For option (d), we have \(\ell = 4\) and \(m = -6\). Here, \(\ell = 4\) is within the range for \(n = 5\). However, the possible values of \(m\) for \(\ell = 4\) are \(-4, -3, -2, -1, 0, 1, 2, 3, 4\). As \(m=-6\) is not within this range, this option is invalid.

Based on the quantum number rules, only option (c) with \(\ell = 3\) and \(m = -2\) is valid for the electron in the hydrogen atom when it is in the \(n = 5\) state.