JEE Advanced 2025 · Paper 1, Physics question 13

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 $k_BT$ has the same de Broglie wavelength as that of this electron. If this temperature is given by $T = \dfrac{Z^2h^2}{\alpha\pi^2a_0^2m_Nk_B}$, (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 $\alpha$ is _____.