Answer

Verified

429.9k+ views

**Hint:**To answer this question, you must recall the quantum mechanical model of an atom given by Irwin Schrodinger. According to Schrodinger, electrons are represented by an amplitude wave function notated by the Greek alphabet psi: $\Psi $.

**Complete step by step answer:**

We know that, at radial nodes, ${\Psi ^2} = 0$

Thus, we can say $\Psi = 0$

Now we are given the following relation in the question $\Psi \left( {{\text{radial}}} \right) = \dfrac{1}{{16\sqrt 4 }}{\left( {\dfrac{Z}{{{a_0}}}} \right)^{3/2}}\left[ {\left( {\sigma - 1} \right)\left( {{\sigma ^2} - 8\sigma + 12} \right)} \right]{e^{ - \sigma /2}}$

Solving for $\Psi = 0$,

We can write $\sigma - 1 = 0$and ${\sigma ^2} - 8\sigma + 12 = 0$

Solving, we get, $\sigma = 1$

Substituting this value into $\sigma = \dfrac{{2Zr}}{{{a_0}}}$, we can find the distance ${r_1}$as, $r = \dfrac{{{a_0}}}{{2Z}}$

Solving the equation, ${\sigma ^2} - 8\sigma + 12 = 0$, we get, $\left( {\sigma - 6} \right)\left( {\sigma - 2} \right) = 0$

Thus, we have $\sigma = 6$ and $\sigma = 2$

The values of $r$ for these values of $\sigma $are ${r_6} = \dfrac{{3{a_0}}}{Z}$and ${r_2} = \dfrac{{{a_0}}}{Z}$respectively.

**Thus, the correct answer is C.**

**Note:**

In the quantum mechanical model of an atom, the electron is believed to be in the form of a wave moving around the nucleus in 3-D space with constant energy. There are some regions around the nucleus where the probability of finding electrons is very high since they are well- defined quantized states that have a minimum possible energy and maximum stability in that region.

When we solve Schrodinger’s equation, the solution gives us the possible energy levels that can be occupied by the electrons and the corresponding wave functions of the electron in each of these energy levels. These quantized energy states and corresponding wave functions are characterized by a set of three quantum numbers ( $n,l,m$). The wave functions arise as a natural consequence in the solution of the Schrodinger equation.

When an electron is present in any energy state, the wave function corresponding to that energy state provides us all the information about that electron. A wave function is a mathematical function that depends on the coordinates of the electron in the atom. It does not carry any physical significance of its own. These wave functions are termed as atomic orbitals.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is BLO What is the full form of BLO class 8 social science CBSE