Answer
a. \( n=4 \) → Maximum number of electrons = 32
b. \( n=5, m_{\ell}=+1 \) → Maximum number of electrons = 8
c. \( n=5, m_{s}=+\frac{1}{2} \) → Maximum number of electrons = 1
d. \( n=3, \ell=2 \) → Maximum number of electrons = 10
e. \( n=2, \ell=1 \) → Maximum number of electrons = 6
Work Step by Step
To determine the maximum number of electrons in an atom with specific quantum numbers, we need to consider the Pauli exclusion principle and the rules for filling electron orbitals.
a. \( n=4 \):
For the principal quantum number \( n=4 \), the maximum value of the azimuthal quantum number \( \ell \) is \( n-1 = 3 \). Therefore, the possible values for \( \ell \) are 0, 1, 2, and 3. Each \( \ell \) value corresponds to a subshell: \( s \) (0), \( p \) (1), \( d \) (2), and \( f \) (3). Each subshell can hold a maximum of \( 2(2\ell+1) \) electrons. So, for \( n=4 \), the maximum number of electrons is:
\( 2(2(0)+1) + 2(2(1)+1) + 2(2(2)+1) + 2(2(3)+1) = 2 + 6 + 10 + 14 = 32 \)
Therefore, the maximum number of electrons in an atom with \( n=4 \) is 32.
b. \( n=5, m_{\ell}=+1 \):
For $n=5$, $l=0,1,2,3,4$. For $l=1,2,3,4$, all can have $m_l=+1$. Four distinct orbitals, thus 8 electrons.
c. \( n=5, m_{s}=+\frac{1}{2} \):
The spin quantum number \( m_{s} \) can have two values: \( +\frac{1}{2} \) (spin-up) and \( -\frac{1}{2} \) (spin-down). Since \( m_{s} = +\frac{1}{2} \), it represents one possible spin state. There are 25 orbitals with $n=5$. They can hold 50 electrons.
d. \( n=3, \ell=2 \):
For \( n=3 \), the possible values for \( \ell \) are 0, 1, and 2. Since \( \ell = 2 \), it represents the \( d \) subshell. The maximum number of electrons in the \( d \) subshell is \( 2(2(2)+1) = 10 \). Therefore, the maximum number of electrons in an atom with \( n=3 \) and \( \ell=2 \) is 10.
e. \( n=2, \ell=1 \):
For \( n=2 \), the possible values for \( \ell \) are 0 and 1. Since \( \ell = 1 \), it represents the \( p \) subshell. The maximum number of electrons in the \( p \) subshell is \( 2(2(1)+1) = 6 \). Therefore, the maximum number of electrons in an atom with \( n=2 \) and \( \ell=1 \) is 6.
