Question

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=0, \ell=0, m_{\ell}=0\) b. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-\frac{1}{2}\) c. \(n=3, m_{s}=+\frac{1}{2}\) d. \(n=2, \ell=2\) e. \(n=1, \ell=0, m_{\ell}=0\)

Short answer

In summary: - a. There are 0 electrons. - b. There is 1 electron. - c. There are 10 electrons. - d. There are 0 electrons. - e. There are 2 electrons.

Step-by-step solution

Check validity of quantum numbers

Let's verify the quantum numbers' validity. Principle quantum number \(n\) should be a positive integer, meaning \(n=1,2,3,...\). Here, \(n=0\), which is not a valid value for the principle quantum number. Thus, the given quantum numbers are invalid.

Number of electrons

Since the given quantum numbers are invalid, there will be no electrons having these quantum numbers. b. \(n=2\), \(\ell=1\), \(m_{\ell}=-1\), \(m_{s}=-\frac{1}{2}\)

Check validity of quantum numbers

The given quantum numbers are valid with \(n=2\), \(\ell=1\), \(m_{\ell}=-1\) and \(m_{s}=-\frac{1}{2}\).

Number of electrons

Since we have valid quantum numbers, there is only one maximum electron (due to the unique combination of quantum numbers) that can have these quantum numbers. c. \(n=3\), \(m_{s}=+\frac{1}{2}\)

Check validity of quantum numbers

Principle quantum number \(n\) is valid. We are not given the values of \(\ell\) and \(m_{\ell}\), so let's find the maximum electrons for \(n=3\).

Number of electrons

Since \(n=3\), the various subshells will be \(0,1,2\). For each subshell: - \(\ell = 0\), there will be one orbital, that can accommodate \(2\) electrons with \(m_s = \frac{1}{2}\). - \(\ell = 1\), there will be three orbitals (\(m_\ell = -1, 0, 1\)), and each orbital can accommodate \(1\) electron with \(m_s = \frac{1}{2}\). So, a total of \(3\) electrons for this subshell. - \(\ell = 2\), there will be five orbitals (\(m_\ell = -2,-1,0,1,2\)), and each orbital can accommodate \(1\) electron with \(m_s = \frac{1}{2}\). So, a total of \(5\) electrons for this subshell. Adding electrons from all subshells, we get \(2+3+5 = 10\) electrons. d. \(n=2\), \(\ell=2\)

Check validity of quantum numbers

Here, the principle quantum number is valid, but the value of \(\ell\) is invalid as the maximum value of \(\ell\) should be \(n-1\). Since \(\ell = n > n-1\) for \(n=2\), the value of \(\ell\) is invalid.

Number of electrons

Since the given quantum numbers are invalid, there will be no electrons having these quantum numbers. e. \(n=1\), \(\ell=0\), \(m_{\ell}=0\)

Check validity of quantum numbers

In this case, the given quantum numbers are valid with \(n=1\), \(\ell=0\) and \(m_{\ell}=0\).

Number of electrons

Since we have valid quantum numbers, there is one orbital associated with these quantum numbers. This orbital can accommodate 2 electrons, one with spin \(+\frac{1}{2}\) and another with spin \(-\frac{1}{2}\). So, there can be a maximum of 2 electrons that can have these quantum numbers. In summary: - a. There are 0 electrons. - b. There is 1 electron. - c. There are 10 electrons. - d. There are 0 electrons. - e. There are 2 electrons.