In the Marvel universe, Hank Pym – the original ant-man – has discovered a new type of atomicspecies where the centripetal force on the electron is equal to nkrZ (the variables have their usualmeaning). If all other postulates of Bohr model are valid for this atom, then find –a. Proportionality between velocity of electron and ‘n’ and ‘Z’b. Proportionality between radius of orbit of electron and ‘n’ and ‘Z’c. Nature of graph of Ln vs rn (Ln = angular momentum of electron in nth orbit and rn = radius ofelectron in nth orbit) for hydrogen atomd. Nature of graph between centripetal force and ‘n’ for hydrogen atom

The question is asking for the relationship between various parameters in a modified version of the Bohr model of the atom.

a. In the Bohr model, the centripetal force is given by mv²/r = kZe²/r². In this new atomic species, the centripetal force is given by nkrZ. Equating these two expressions, we get mv² = nkrZ. The velocity of the electron (v) is therefore proportional to the square root of nZ.

b. In the Bohr model, the radius of the electron's orbit is given by n²h²/4π²mkZ. In this new atomic species, the centripetal force is given by nkrZ. Equating these two expressions, we get n²h²/4π²m = r. The radius of the electron's orbit (r) is therefore proportional to n².

c. The angular momentum of the electron in the nth orbit (Ln) is given by mvr = nh/2π. Substituting the expression for r from part b, we get Ln = nh/2π * n²h²/4π²m = n³h³/8π³m. Taking the natural logarithm of both sides, we get ln(Ln) = ln(n³h³/8π³m) = 3ln(n) + ln(h³/8π³m). This is a straight line equation of the form y = mx + c, where y = ln(Ln), x = ln(n), m = 3, and c = ln(h³/8π³m). Therefore, the graph of ln(Ln) vs ln(n) is a straight line with a slope of 3.

d. The centripetal force in this new atomic species is given by nkrZ. For a hydrogen atom, Z = 1. Therefore, the centripetal force is proportional to n. The graph of centripetal force vs n is therefore a straight line with a slope equal to kr.