Question

The electron in a certain hydrogen atom is in the \(n=5\) state. Which of the following could be the \(\ell\) and \(m\) values for the electron? a) 5,-3 b) 4,-5 c) 3,-2 d) 4,-6

Short answer

Answer: c) 3, -2

Step-by-step solution

Identify the given Quantum Number and the possible values of \(\ell\) and \(m\)

We are given that the electron in the hydrogen atom is in the \(n=5\) state. Using the rules for quantum numbers, we can find the possible values of \(\ell\) and \(m\) for the electron. For n=5, possible \(\ell\) values are 0, 1, 2, 3, and 4 because \(0 \leq \ell \leq n-1\). For each possible value of \(\ell\), we can find the possible values of \(m\) using the rule \(-\ell \leq m \leq +\ell\).

Check each option to see if it falls within the possible values for \(\ell\) and \(m\)

We will check each given option and see if they follow the rules for the quantum numbers. a) 5, -3 \(\ell=5\) is not a possible value for \(\ell\) since 5 is not within the range of 0 to n-1. So, this option is incorrect. b) 4, -5 \(\ell=4\) is a possible value for \(\ell\). However, for \(\ell=4\), \(m\) can only range from -4 to +4. Since -5 is outside this range, this option is incorrect. c) 3, -2 \(\ell=3\) is a possible value for \(\ell\), and for \(\ell=3\), \(m\) can range from -3 to +3. Since -2 is within this range, this option is correct. d) 4, -6 \(\ell=4\) is a possible value for \(\ell\), but for \(\ell=4\), \(m\) can only range from -4 to +4. Since -6 is outside this range, this option is incorrect. So, the correct answer is c) 3, -2, as these \(\ell\) and \(m\) values follow the rules for the quantum numbers for the given principal quantum number (n=5).