To determine the number of electrons with a spin quantum number of $$s=+\frac{1}{2}$$ in all completely filled subshells with principal quantum number $$n=4$$, we must first identify the subshells in the n=4 shell and then calculate the electrons with the specified spin.

The n=4 shell has the following subshells and their capacity for electrons:

• $$4s$$ subshell can hold 2 electrons
• $$4p$$ subshell can hold 6 electrons
• $$4d$$ subshell can hold 10 electrons
• $$4f$$ subshell can hold 14 electrons

In quantum mechanics, each orbital within these subshells can hold 2 electrons, each with opposite spins ($$+\frac{1}{2}$$ and $$-\frac{1}{2}$$). Thus, in a completely filled subshell, half of the electrons will have a spin quantum number of $$+\frac{1}{2}$$:

• In $$4s$$, 1 out of 2 electrons will have $$s=+\frac{1}{2}$$
• In $$4p$$, 3 out of 6 electrons will have $$s=+\frac{1}{2}$$
• In $$4d$$, 5 out of 10 electrons will have $$s=+\frac{1}{2}$$
• In $$4f$$, 7 out of 14 electrons will have $$s=+\frac{1}{2}$$

Adding these together: $$1 + 3 + 5 + 7 = 16$$ electrons with $$s=+\frac{1}{2}$$ in all completely filled subshells with $$n=4$$.