In a hydrogen atom, if electron is replaced by a particle which is 40 times heavier but has the same charge, then, the ratio of the radius of the first excited state of a normal hydrogen atom to the ground state of the above atom is

Step-by-step solution

The radius of the n -th orbit of a hydrogen-like atom is given by r_n = n^2 a₀ Z m_e m , where m is the mass of the particle and a₀ is the Bohr radius of the hydrogen atom ( a₀ = h^2 4 ^2 m_e e^2 ). For a normal hydrogen atom, m = m_e , Z = 1 , and the first excited state corresponds to n = 2 . Thus, r_ H = 2^2 a₀ 1 = 4 a₀ . For the new atom, the particle mass is m' = 40 m_e , Z = 1 , and the ground state corresponds to n = 1 . Thus, r' = 1^2 1 m_e 40 m_e a₀ = a₀ 40 . The ratio of the radius of the first excited st