Chemistry - Atomic Structure Question with Solution | TestHub

Question Explanation:

Based on schordinger equation which statements are correct.

Concept:

Schordinger wave equation.

Solution:

The radial part of the wave function for a hydrogenic atom is generally given by $R_{n,l}(r)\propto {\left(\frac{2Zr}{n{a}_n}\right)}^1\times ($ polynomial in $r)\times e^{-Zr/n{a}_0}$.

The given wave function has a term $\sigma {\mathrm{e}}^{-\sigma /2}$, which corresponds to the ${\mathrm{r}}^{\mathrm{l}}{\mathrm{e}}^{-{\mathrm{Zr}}_{\mathrm{r}}/{\mathrm{na}}_0}$ part. The term $\sigma$ can be written as $\frac{2\mathrm{Zr}}{{\mathrm{na}}_0}$. The power of $\sigma$ is 1 , which means $\mathrm{l}=1$.

The polynomial part of the wave function is $(1-\sigma )\left(12-8\sigma +{\sigma }^2\right)$. The number of radial nodes is the number of times the radial wave function changes sign, which is determined by the roots of this polynomial.

Let's find the roots of the polynomial: $(1-\sigma )\left(12-8\sigma +{\sigma }^2\right)=0$

The roots are $\sigma =1$ and the roots of ${\sigma }^2-8\sigma +12=0$.

Factoring the quadratic equation: $(\sigma -6)(\sigma -2)=0$

The roots are $\sigma =2$ and $\sigma =6$.

So, the radial part of the wave function has three roots where it changes sign ( $\sigma =1,2,6$ ). This means there are 3 radial nodes.

The number of radial nodes is also given by the formula $\mathrm{n}-1-1$.

$\mathrm{n}-\mathrm{l}-1=3$

Since $\mathrm{l}=1$, we have $\mathrm{n}-1-1=3$, which gives $\mathrm{n}-2=3$, so $\mathrm{n}=5$. Therefore, the orbital is a 5 p orbital.

Statement (i): It has 2 radial nodes.

The number of radial nodes is $n-1-1=5-1-1=3$. The statement says 2 radial nodes. This statement is incorrect.

Statement (ii): It has 1 angular node.

The number of angular nodes is equal to the azimuthal quantum number l . For a 5 p orbital, $\mathrm{l}=1$. So, there is 1 angular node. This statement is correct.

Statement (iii): The total number of nodal surfaces are 4.

The total number of nodes is the sum of radial nodes and angular nodes.

Total nodes $=($ Radial nodes $)+($ Angular nodes $)=(n-1-1)+1=n-1$.

For a 5p orbital, total nodes $=5-1=4$. This statement is correct.

Statement (iv): The radial wave function changes its sign three times in $\psi (r)$ vs $r$ curve.

The number of sign changes in the radial wave function corresponds to the number of radial nodes. As determined in Step 1, the number of radial nodes is 3 . This statement is correct. Statement (v): The given orbital may be 6pz .

The orbital is a 5p orbital, not a 6p orbital. This statement is incorrect.

Final Answer: 3.00