To determine how many electrons can occupy the given orbitals and energy levels of an atom, it's important to understand the rules of electron configuration and the capacities of each type of orbital.
1. 1s orbital:
The 's' orbital can hold a maximum of 2 electrons. Therefore:
[tex]\[
\text{Number of electrons in } 1s = 2
\][/tex]
2. 6d_{x^2-y^2} orbital:
The 'd' subshell consists of 5 orbitals, each of which can hold a maximum of 2 electrons. Specifically, the [tex]\( d_{x^2-y^2} \)[/tex] orbital is one of these 5 d orbitals. Therefore, each [tex]\( d \)[/tex] orbital, including [tex]\( 6d_{x^2-y^2} \)[/tex], can hold up to 2 electrons:
[tex]\[
\text{Number of electrons in } 6d_{x^2-y^2} = 2
\][/tex]
3. 4f orbital:
The 'f' subshell consists of 7 orbitals, each of which can hold a maximum of 2 electrons. Thus, the total number of electrons that can fit into the 'f' subshell (such as 4f) is:
[tex]\[
\text{Number of electrons in } 4f = 14
\][/tex]
4. 7p_y orbital:
The 'p' subshell consists of 3 orbitals (p_x, p_y, p_z), each of which can hold a maximum of 2 electrons. Therefore, any individual 'p' orbital, such as [tex]\( 7p_y \)[/tex], can hold:
[tex]\[
\text{Number of electrons in } 7p_y = 2
\][/tex]
5. 2s orbital:
Similar to the 1s orbital, the 2s orbital can hold a maximum of 2 electrons, since it is an 's' orbital:
[tex]\[
\text{Number of electrons in } 2s = 2
\][/tex]
6. n=1 energy level:
The principal quantum number n=1 consists only of the 1s orbital, which can hold a maximum of 2 electrons. Hence, the total number of electrons that can occupy the n=1 energy level is:
[tex]\[
\text{Number of electrons in } n=1 = 2
\][/tex]
Putting all of these together:
- [tex]\( 1s \)[/tex]: 2 electrons
- [tex]\( 6d_{x^2-y^2} \)[/tex]: 2 electrons
- [tex]\( 4f \)[/tex]: 14 electrons
- [tex]\( 7p_y \)[/tex]: 2 electrons
- [tex]\( 2s \)[/tex]: 2 electrons
- [tex]\( n=1 \)[/tex]: 2 electrons
