Question

the hypothesis that was put forward by louis de broglie in 1924 was astonishing for a number of reasons. an obvious reason is that associating a wavelike nature with particles is far from intuitive, but another astonishing aspect was how well the hypothesis fit in with certain parts of existing physics. in this problem, we explore the correspondence between the de broglie picture of the wave nature of electrons and the bohr model of the hydrogen atom. what i sthe circumference c₃ of the third (n=3) energy level in hydrogen predicted by the Bohr model of the atom?

Answer 1

The circumference of the third energy level in the hydrogen atom as predicted by Bohr's model is calculated using the quantization of angular momentum and the de Broglie wavelength for an electron in orbit.

Step-by-step explanation:

The question asks for the circumference of the third energy level (n=3) in the hydrogen atom as predicted by Bohr's model. According to Bohr's model, the circumference of an electron orbit must be an integral multiple of the de Broglie wavelength. The de Broglie wavelength, λ, is given by the equation λ = h/mv, where 'h' is Planck's constant, 'm' is the mass of the electron, and 'v' is the velocity of the electron.

The quantization of angular momentum in Bohr's model (L = n∙h/2π) implies that n∙λ = 2πr, where 'r' is the radius of the n-th orbit and 'n' is the principal quantum number. Using the fact that the radius of the nth orbit in the Bohr model is given by r_n = n^2 ∙ (h^2/4π^2 ∙ (mke^2), we can calculate the circumference (C = 2πr). For n=3, the circumference, C_3, of the third energy level is C_3 = 2πr_3.