The number of electrons present in all the completely filled subshells having $\mathrm{n}=4$ and $\mathrm{s}=+\frac{1}{2}$ is _______.

(Where $\mathrm{n}=$ principal quantum number and $\mathrm{s}=$ spin quantum number)

<p>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.</p> <p>The n=4 shell has the following subshells and their capacity for electrons:</p> <ul> <li>$4s$ subshell can hold 2 electrons</li> <li>$4p$ subshell can hold 6 electrons</li> <li>$4d$ subshell can hold 10 electrons</li> <li>$4f$ subshell can hold 14 electrons</li> </ul> <p>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 $+\frac{1}{2}$:</p> <ul> <li>In $4s$, 1 out of 2 electrons will have $s=+\frac{1}{2}$</li> <li>In $4p$, 3 out of 6 electrons will have $s=+\frac{1}{2}$</li> <li>In $4d$, 5 out of 10 electrons will have $s=+\frac{1}{2}$</li> <li>In $4f$, 7 out of 14 electrons will have $s=+\frac{1}{2}$</li> </ul> <p>Adding these together: $1 + 3 + 5 + 7 = 16$ electrons with $s=+\frac{1}{2}$ in all completely filled subshells with $n=4$.</p>