A hypothetical atom consists of a particle of mass m moving in a circular orbit around a central stationary nucleus. The potential energy of the particle in this field is given by U(r) = r^4 , where is a positive constant and r is the distance from the center. If Bohr's quantization postulate is applied to this system, how do the radius of the n^ th orbit ( r_n ) and the kinetic energy of the particle in that orbit (

Step-by-step solution

The potential energy of the particle is given by U(r) = r^4 . The conservative force acting on the particle is F = - dU dr = -4 r^3 . The magnitude of this force provides the necessary centripetal force for circular motion: mv^2 r = 4 r^3 mv^2 = 4 r^4 v = 4 m r^2 According to Bohr's quantization postulate, the angular momentum is quantized: mvr = nh 2 Substituting the expression for v into the quantization condition: m ( 4 m r^2 ) r = nh 2 4 m r^3 = nh 2 r^3 = nh 2 4 m From this, we can see that the radius of the n