To determine which set of quantum numbers represents an impossible arrangement, we need to understand the rules governing the values of these quantum numbers:

1. Principal Quantum Number ($n$): Represents the shell (energy level) of an electron in an atom and can take positive integer values such that $n = 1, 2, 3, \ldots$
2. Azimuthal (or Angular Momentum) Quantum Number ($l$): Defines the shape of the orbital, and for any given $n$, $l$ can take values from $0$ to $n - 1$. For example, if $n = 3$, then $l$ can be $0$, $1$, or $2$.
3. Magnetic Quantum Number ($m_{l}$): Describes the orientation of the orbital in space relative to the other orbitals and can have integer values ranging from $-l$ to $+l$, including zero. Thus, for $l = 2$, $m_{l}$ can be $-2, -1, 0, 1, 2$.
4. Spin Quantum Number ($m_{s}$): Specifies the spin of the electron and can take values of $-1/2$ or $+1/2$.

Now, let's evaluate the given options:

Option A: $n = 3, l = 2, m_{l} = -2, m_{s} = 1/2$

This set of quantum numbers is possible because:

• With $n = 3$, $l$ can be $0$, $1$, or $2$, so $l = 2$ is valid.


• For $l = 2$, $m_{l}$ can range from $-2$ to $2$, so $m_{l} = -2$ is valid.


• $m_{s}$ has a valid value of $1/2$.

Option B: $n = 4, l = 0, m_{l} = 0, m_{s} = 1/2$

This combination is also correct because:

• With $n = 4$, $l$ can be $0, 1, 2, 3$, hence $l = 0$ is valid.


• For $l = 0$, the only valid value for $m_{l}$ is $0$, which matches.


• $m_{s} = 1/2$ is a valid value for the spin quantum number.

Option C: $n = 3, l = 2, m_{l} = -3, m_{s} = 1/2$

This set of quantum numbers is impossible because:

• Although $n = 3$ and $l = 2$ are valid combinations, for $l = 2$, $m_{l}$ can only be $-2, -1, 0, 1,$ or $2$. The value of $m_{l} = -3$ is outside this permissible range, making this set of quantum numbers impossible.

Option D: $n = 5, l = 3, m_{l} = 0, m_{s} = -1/2$

This option is also possible because:

• With $n = 5$, $l = 3$ is within the valid range ($0$ through $4$).


• For $l = 3$, $m_{l} = 0$ falls within the valid range ($-3, -2, -1, 0, 1, 2, 3$).


• $m_{s} = -1/2$ is a valid spin quantum number value.

Therefore, the impossible arrangement among the given options is Option C: $n = 3, l = 2, m_{l} = -3, m_{s} = 1/2$.