JEE Advanced Atoms Previous Year Questions (PYQ) with Detailed Solutions

Q 1 :

Consider an electron in the $n = 3$ orbit of a hydrogen-like atom with atomic number $Z$ . At absolute temperature $T$ , a neutron having thermal energy $k_{B} T$ has the same de Broglie wavelength as that of this electron. If this temperature is given by $T = \frac{z^{2} h^{2}}{α π^{2} a_{0}^{2} m_{N} k_{B}},$ (where $h$ is the Planck's constant, $k_{B}$ is the Boltzmann constant, $m_{N}$ is the mass of the neutron and $a_{0}$ is the first Bohr radius of hydrogen atom) then the value of $α$ is ______. [2025]

(72)

From Bohr's postulates,

$m v r = \frac{n h}{2 π}$ $\dots (i)$

and $\frac{m v^{2}}{r} = \frac{K Z e^{2}}{r^{2}}$

$\Rightarrow m v^{2} r = \frac{1}{4 π ε_{0}} Z e^{2}$ $\dots (i i)$

Dividing Eq. (ii) by Eq. (i),

$v = \frac{\frac{Z e^{2}}{4 π ε_{0}}}{\frac{n h}{2 π}} = \frac{Z e^{2}}{2 ε_{0} n h}$

$λ = \frac{h}{m v} = \frac{h}{\sqrt{2 m_{N} K_{B} T}}$

Now putting the above value of $v$ and solving, we get

$T = \frac{m^{2} Z^{2} e^{4}}{8 ε_{0}^{2} n^{2} h^{2} m_{N} K_{B}}$ or, $T = \frac{m^{2} Z^{2} e^{4}}{72 ε_{0}^{2} h^{2} m_{N} K_{B}}$ $[∵ n = 3 given]$

Again dividing Eq. (ii) by Eq. (i), we get

$\frac{1}{m r} = \frac{\frac{Z e^{2}}{4 π ε_{0}}}{\frac{n^{2} h^{2}}{4 π^{2}}}$

$r = \frac{n^{2} h^{2} ε_{0}}{π Z e^{2} m}$ $\Rightarrow a_{0} = \frac{h^{2} ε_{0}}{π e^{2} m}$ $[∵ r = a_{0} n^{2}]$

or, $a_{0}^{2} = \frac{h^{4} ε_{0}^{2}}{π^{2} e^{4} m^{2}}$

$T a_{0}^{2} = \frac{m^{2} Z^{2} e^{4}}{72 ε_{0}^{2} h^{2} m_{N} K_{B}} \cdot \frac{h^{4} ε_{0}^{2}}{π^{2} e^{4} m^{2}}$

or, $T = \frac{h^{2} Z^{2}}{72 π^{2} a_{0}^{2} m_{N} K_{B}}$

Comparing with the given $T = \frac{Z^{2} h^{2}}{α π^{2} a_{0}^{2} m_{N} K_{B}},$

we get $α = 72$

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