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= Z^2 h^2 ^2 a₀^2 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 a₀ is the first Bohr radius of hydrogen atom

Step-by-step solution

mv ^2 r = KZe ^2 r ^2 mv ^2 r = 1 4 ₀ Ze ^2 mvr = nh 2 (1)/(2) gives v = Ze ^2 4 ₀ nh 2 = Ze ^2 2 ₀ nh aligned & h m v = h 2 m_N K_B T & T= m^2 Z^2 e^4 8 ₀^2 n^2 h^2 m_N K_B & n=3 T= m^2 Z^2 e^4 72 ₀^2 h^2 m_N K_B & (1) (2)^2 1 m r = Z e^2 4 ₀ n^2 h^2 4 ^2 aligned aligned & r = n ^2 ~h ^2 ₀ Ze ^2 ~m a ₀= h ^2 ₀ e ^2 ~m & a ₀^2= h ^4 ₀^2 ^2 e ^4 ~m ^2 & Ta ₀^2= m ^2 Z ^2 e ^4 72 ₀ ~h ^2 ~m _ N k _ B h ^4 ₀^2 ^2 e ^4 ~m ^2 & ~T = h ^2 Z ^2 72 ^2 a ₀^2 ~m _ N k _ B =72 aligned