What is the maximum number of electron states in the \(n=5\) shell?

For a given principal quantum number (n), the azimuthal quantum number (l) can have values from 0 to n-1, inclusive. In this case, n=5, so the allowed values for l are {0, 1, 2, 3, 4}.

For each value of azimuthal quantum number (l), the magnetic quantum number (m_l) can have values from -l to +l, inclusive. Now let's identify the allowed values for m_l for each value of l found in Step 1: - For l=0: m_l = {0} - For l=1: m_l = {-1, 0, 1} - For l=2: m_l = {-2, -1, 0, 1, 2} - For l=3: m_l = {-3, -2, -1, 0, 1, 2, 3} - For l=4: m_l = {-4, -3, -2, -1, 0, 1, 2, 3, 4}

The spin quantum number (m_s) can have two possible values for each electron: +1/2 and -1/2.

To find the maximum number of electron states, we need to count the number of allowed combinations of l, m_l, and m_s for n=5. This can be done by taking the product of the number of allowed values for each l and m_l combination, multiplied by 2 (since there are two possible values for m_s): - For l=0 and m_l=0: 1 * 2 = 2 electron states - For l=1 and m_l=(-1, 0, 1): 3 * 2 = 6 electron states - For l=2 and m_l=(-2, -1, 0, 1, 2): 5 * 2 = 10 electron states - For l=3 and m_l=(-3, -2, -1, 0, 1, 2, 3): 7 * 2 = 14 electron states - For l=4 and m_l=(-4, -3, -2, -1, 0, 1, 2, 3, 4): 9 * 2 = 18 electron states Now, add together the number of electron states for each l value: 2 + 6 + 10 + 14 + 18 = 50 So, the maximum number of electron states in the n=5 shell is 50.