Question

please give me the step and answer!

This graph shows two curves pertaining to a hydrogen s orbital. What is the value of the principal quantum number, n, for the hydrogen s orbital matching these curves? How many nodes are predicted by this figure? According to the figure, select the appropriate x-axis values for the location of nodes in terms of the Bohr radii. Here is an expanded view of just the portion of the graph from x = 0 to x = 8.

Answer 1

General guidance

Concepts and reason

The concept that is used, in order to solve the given problem is the cross cutting of the ${\\rm{X}}$-axis by the radial wave function. The cross cutting of the ${\\rm{X}}$-axis by the radial wave function determines the number of nodes, which can be further employed in the determination of the principal quantum number.

Fundamentals

Principal quantum number for any electron describe its state. Radial probability wave function describes the region in which the probability of finding an electron would be highest. Furthermore, principal quantum number can be find out using the below mentioned formula:

${\\rm{Principal quantum number }}\\left( n \\right)\\; = \\;{\\rm{Number of nodes }} + {\\rm{ }}1$

Step-by-step

Step 1 of 3

(1)

The principal quantum number can be given by the below mentioned formula:

${\\rm{Principal quantum number }}\\left( n \\right)\\; = \\;{\\rm{Number of nodes }} + {\\rm{ }}1$

Substitute, 4 for number of nodes,

$\\begin{array}{c}\\\\{\\rm{Principal quantum number }}\\left( n \\right){\\rm{ }} = {\\rm{ }}4 + 1\\\\\\\\ = \\;5\\\\\\end{array}$

The principal quantum number of the hydrogen \u2018$s$\u2019 orbital represented in the graph is $5$.

Explanation | Common mistakes | Hint for next step

The principal quantum number and nodes number are related to each other. The number of nodes is given by the number of times the radial wave function cuts the${\\rm{X}} - {\\rm{axis}}$. In the given graph, the radial wave function cuts the ${\\rm{X}} - {\\rm{axis}}$ $4$ times.

Hence the value for the number of nodes is equal to$4$. Further, the orbital \u2018$ns$\u2019 is considered to have $\\left( {n - 1} \\right)$radial nodes. As a result of which $n - 1$will signify the principal quantum number.

Step 2 of 3

(2)

There a total of two curves present in the given figure, radial probability density and radial wave function. The total number of nodes that can be predicted with the help of the given figure is$4$.

The total number of nodes that can be predicted from the graph is$4$.

Explanation | Hint for next step

The number of nodes is related to the radial wave function curve represented in a graph. In the given figure, the radial wave function cross cuts the ${\\rm{X}} - {\\rm{axis}}$ $4$ times, hence the value of the number of nodes is equal to$4$.

Step 3 of 3

(3)

The radial wave function curve is represented by the orange curve in the figure given in the problem. It further cuts the ${\\rm{X}} - {\\rm{axis}}$at $4$ different points which are $1.9$, $6.4$, $13.9$and $27$.

The values of ${\\rm{X}} - {\\rm{axis}}$ for the location of nodes are$1.9$, $6.4$, $13.9$ and $27$.

The values for the location of nodes on the ${\\rm{X}} - {\\rm{axis}}$is to be selected by choosing the points, where the curve cuts the ${\\rm{X}} - {\\rm{axis}}$. The number of nodes is represented by the cuts made by the radial wave function curve. Hence, those points will be the appropriate ${\\rm{X}} - {\\rm{axis}}$values.

Answer

The principal quantum number of the hydrogen \u2018$s$\u2019 orbital represented in the graph is $5$.

The total number of nodes that can be predicted from the graph is$4$.

The values of ${\\rm{X}} - {\\rm{axis}}$ for the location of nodes are$1.9$, $6.4$, $13.9$ and $27$.

Answer only

The principal quantum number of the hydrogen \u2018$s$\u2019 orbital represented in the graph is $5$.

The total number of nodes that can be predicted from the graph is$4$.

The values of ${\\rm{X}} - {\\rm{axis}}$ for the location of nodes are$1.9$, $6.4$, $13.9$ and $27$.