Question

The wave function, ${\psi }_{n,l,m_{\ell }}$ is a mathematical function whose value depends upon spherical polar coordinates ($r,\theta ,\varphi$ ) of the electron and characterized by the quantum numbers $n,\ell$ and $m_l.$ Here $r$ is distance from nucleus, $\theta$ is colatitude and $\varphi$ is azimuth. In the mathematical functions given in the Table, $Z$ is atomic number and $a_0$ is Bohr radius.

Column 1	Column 2	Column 3
(I) Is orbital	(i) ${\psi }_{n,l,m_t}\propto {\left(\frac{Z}{a_e}\right)}^{\frac{3}{2}}{e}^{-\left(\frac{z}{a_0}\right)}$	(p)
(II) 2s orbital	(ii) one radial node	(Q) Probability density at nucleus $\propto \frac{1}{a_0^3}$
(III) $2p_z$ orbital	(iii) ${\psi }_{n,l,m_i}\propto {\left(\frac{Z}{a_o}\right)}^{\frac{5}{2}}r{e}^{-\left(\frac{z}{2a_0}\right)}\cos \theta$	(R) Probability density is maximum at nucleus
(IV) $3d_z^2$ Orbital	(iv) xy-plane is a nodal plane	(S) Energy needed to excite electron from $n=2$ state to $n=4$ state is $\frac{27}{32}$ times the energy needed to excite electron from $n=2$ state to $n=6$ state

For $H{e}^{+}$ ion, the only INCORRECT combination is

• A. (I) (i) (S)
• B. (II) (i) (Q)
• C. (I) (i) (P)
• D. (I) (iv) (R)

Answer 1

Answer: (A)

$1s\to {\psi }_{n,\hat{\ell },m_{\ell }}\propto {\left(\frac{Z}{a_0}\right)}^{\frac{3}{2}}{e}^{-\left(\frac{Zr}{a_0}\right)}$
$2s\to$ One Radial Node
$2p_z\to xy$ plane is a Nodal plane
$3d_{z^2}\to {\psi }_{n,\ell ,m_{\ell }}\propto {\left(\frac{Z}{a_0}\right)}^{\frac{5}{2}}r{e}^{-\left(\frac{Zr}{2a_0}\right)}\cos \theta$