Question

Which orbital notation does not have spherical node______.
a. $n = 2, l = 0$
b. $n = 2, l = 1$
c. $n = 3, l = 0$
d. $n = 3, l = 1$

Answer 1

Hint: Spherical node is also called a Radial node. The spherical node occurs when the spherical wave function of an atomic orbital is zero. To get the answer, use the formula that relates the principal quantum number and azimuthal quantum number.

Formula used:
In an orbit, a spherical node can be calculated by
$ \Rightarrow n - l - 1$.
Where $n$ is the principal quantum number and $l$ is the angular momentum number.

Complete step by step answer:
In the question, it is to find the orbital notation which does not have a spherical node. First, let us know the meaning of the spherical node. The spherical node is also known as a radial node that can be calculated as $n - l - 1$. Where $n$ is the principal quantum number and $l$ is the angular momentum number. $l$ is known as the angular quantum number that can be described as the shape of an electron. The angular quantum number can be calculated by the formula $n - 1$, where $n$ is $1$, $2$ for $p$, $3$ for $d$. $l$ is for $0$ for $s$, $1$ for $p$, $2$ for $d$. And so on.

Now let us try to answer the given question. Let us consider all the given options.
Consider the option(A). We have $n = 2;l = 0$. A number of spherical nodes by the formula $n - l - 1$. Consider,
$ \Rightarrow n - l - 1$
Let us substitute the values.
$ \Rightarrow 2 - 0 - 1$
On subtracting we get,
$ \Rightarrow 1$
Therefore $2s$ have $1$ spherical nodes.

Consider, option (B). we have, $n = 2;l = 1$. The number of spherical nodes by the formula $n - l - 1$. Consider,
$ \Rightarrow n - l - 1$
Let us substitute the values.
$ \Rightarrow 2 - 1 - 1$
On subtracting we get,
$ \Rightarrow 0$
$ \Rightarrow 2p$ has no spherical node.

Consider the option(C). we have, $n = 3;l = 0$. The number of spherical nodes by the formula $n - l - 1$. Consider,
$ \Rightarrow n - l - 1$
Let us substitute the values.
$ \Rightarrow 3 - 0 - 1$
On subtracting we get,
$ \Rightarrow 2$
Therefore $3s$ have $2$ spherical nodes.

Consider the option(D). we have,$n = 3;l = 1$. The number of spherical nodes by the formula $n - l - 1$. Consider,
$ \Rightarrow n - l - 1$
Let us substitute the values.
$ \Rightarrow 3 - 1 - 1$
On subtracting we get,
$ \Rightarrow 1$
Therefore $2s$ have $1$ a spherical node.

Hence, the correct answer is option (B).

Note: We have used this formula $n - l - 1$ throughout the question to get the answer. Do not confuse between $n - l - 1$ and $l = n - 1$. The meaning of $n$ is different in both the formulae. In the first formula, $n$ is the number of shells and in the second formula $n$ is the principal quantum number.