Question

which of the following is not a valid set of quantum numbers? n = 3, l = 3, m = 3; n = 1, l = 0, m = 0; n = 2, l = 1, m = 0. which of the following gives the correct possible values of l for n = 4? 0, 1, 2, 3; -4, -3, -2, -1, 0, 1, 2, 3, 4; 0, 1, 2, 3, 4

Explanation:

1. Recall the rules for quantum - numbers:

• Principal quantum number ($n$): $n = 1,2,3,\cdots$. It represents the energy level or shell of an electron.
• Azimuthal quantum number ($l$): For a given $n$, $l=0,1,\cdots,n - 1$. It determines the shape of the orbital.
• Magnetic quantum number ($m$): For a given $l$, $m=-l,-l + 1,\cdots,0,\cdots,l-1,l$.

1. Analyze the first part (checking valid set of quantum - numbers):

• For $n = 3$, the possible values of $l$ are $l = 0,1,2$ (since $l$ ranges from $0$ to $n - 1$). Here, $l = 3$ is not allowed when $n = 3$. So the set $n = 3,l = 3,m = 3$ is not a valid set of quantum - numbers.
• For $n = 1$, $l$ can only be $0$ (because $n-1=0$ when $n = 1$), and when $l = 0$, $m = 0$ is valid.
• For $n = 2$, $l$ can be $0$ or $1$, and when $l = 1$, $m = 0$ is a valid value.

1. Analyze the second part (finding possible values of $l$ for $n = 4$):

• Given $n = 4$, the possible values of $l$ range from $0$ to $n - 1$. So $l=0,1,2,3$.
• The set $0,1,2,3$ is the correct set of possible values of $l$ for $n = 4$. The set $-4,-3,-2,-1,0,1,2,3,4$ is incorrect because $l$ cannot be negative. The set $0,1,2,3,4$ is incorrect because $l$ cannot be equal to $n$.

Step1: Recall quantum - number rules

The principal quantum number $n$ is a positive integer. The azimuthal quantum number $l$ ranges from $0$ to $n - 1$, and the magnetic quantum number $m$ ranges from $-l$ to $l$.

Step2: Analyze first set of quantum - numbers

For $n = 3$, $l$ cannot be $3$ as $l$ should satisfy $l\in\{0,1,2\}$. The other two sets ($n = 1,l = 0,m = 0$ and $n = 2,l = 1,m = 0$) are valid.

Step3: Analyze possible $l$ values for $n = 4$

Since $l$ ranges from $0$ to $n - 1$ for a given $n$, when $n = 4$, $l = 0,1,2,3$.

Answer:

The set $n = 3,l = 3,m = 3$ is not a valid set of quantum - numbers. The correct possible values of $l$ for $n = 4$ are $0,1,2,3$.