Two electrons are moving in the 3^ rd and 4^ th orbits of two different hydrogen-like atoms. If the ratio of the radii of these orbits is 9 : 8 , the ratio of the speeds of the electrons in these orbits is:

Step-by-step solution

For hydrogen-like atoms, the radius of the n^ th orbit is r n^2 Z and the speed of the electron is v Z n . From the speed relation, we can write Z n v . Substituting this into the radius relation gives: r n^2 n v = n v Rearranging for speed, we get v n r . Given n₁ = 3 , n₂ = 4 and r₁ r₂ = 9 8 , the ratio of speeds is: v₁ v₂ = n₁ n₂ r₂ r₁ = 3 4 8 9 = 2 3 Thus, the ratio of their speeds is 2 : 3 . Answer: 2 : 3