1   Choose the correct alternative from the clues given at the end of the each statement:$\\$ (a) The size of the atom in Thomson's model is -------- the atomic size in Rutherford's model. (much greater than/no different from/much than.)$\\$ (b) In the ground state of ------------ electrons are in stable equilibrium, while in ....... electrons always experience a net force. (Thomson's model/ Rutherford's model.)$\\$ (c) A classical atom based on ---------- is doomed to collapse. (Thomson's model/ Rutherford's model.)$\\$ (d) An atom has a nearly continuous mass distribution in a--------- but has a highly non- uniform mass distribution in _____ (Thomson's model/ Rutherford's model.)$\\$ (e) The positively charged part of the atom possesses most of the mass in --------- (Rutherford's model/both the models.)

Solution :

(a) The sizes of the atoms taken in Thomson's model and Rutherford's model have the $\underline{same \ order \ of \ magnitude.}$$\\$ (b) In the ground state of $\underline{Thomson's \ model}$, the electrons are in stable equilibrium. However, in Rutherford's model, the electrons always experience a net force.$\\$ (c) A classical atom based on $\underline{Rutherford's \ model}$ is doomed to collapse.$\\$ (d) An atom has a nearly continuous mass distribution in $\underline{Thomson's \ model}$, but has a highly non-uniform mass distribution in ${Rutherford's \ model.}$$\\$ (e) The positively charged part of the atom possesses most of the mass in $\underline{both the models.}$

2   Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet of solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below $14 K.$) What results do you expect?

Solution :

In the alpha-particle scattering experiment, if a thin sheet of solid hydrogen is used in place of a gold foil, then the scattering angle would not be large enough. This is because the mass of hydrogen is less than the mass of incident $\alpha $ - particles Thus, the mass of the scattering particle is more than the target nucleus (hydrogen). As a result, the $\alpha$- particles would not bounce back if solid hydrogen is used in the $\alpha -$ particle scattering experiment and so we cannot determine size of the hydrogen

3   What is the shortest wavelength present in the Paschen series of spectral lines?

Solution :

Rydberg’s Formula is given as:$\\$ $\dfrac{hc}{\lambda} =21.76*10^{-19}\left[\dfrac{1}{n_1^2} -\dfrac{1}{n_2^2}\right]$$\\$ Where, $h =$ Planck’s constant $= 6.6*10^{-34} Js$$\\$ $c=$ Speed of light =$3*10^8 m/s$$\\$ ($ n _1$ and $n _2$ are integers)$\\$ The shortest wavelength present in the Paschen series of the spectral lines is for values $n_1 = 3$ and $n_2 =\infty$$\\$ $\dfrac{hc}{\lambda}=21.76*10^{-19}\left[\dfrac{1}{(3)^2}-\dfrac{1}{(\infty)^2}\right]$$\\$ $\lambda =\dfrac{6.6*10^{-34}*3*10^8*9}{21.76*10^{-19}} \\ =8.189*10^{-7}m\\ 818.9 nm$

4   A difference of $2.3 eV $ separates two energy levels in an atom. What is the frequency of radiation emitted when the atom makes a transition from the upper level to the lower level?

Solution :

Separation of two energy levels in an atom,$\\$ $E=2.3eV\\ =2.3*1.6*10^{-19}\\ =3.68*10^{-19}J$$\\$ Let $\theta $ be the frequency of radiation emitted when the atom transits from the upper level to the lower level. $\\$ We have the relation for energy as:$\\$ $E=hv$$\\$ Where,$\\$ $h=$ Planck's constant$=6.62*10^{-34} Js$$\\$ $\therefore v=\dfrac{E}{h}\\ =\dfrac{3.68*10^{-19}}{6.62*10^{-32}}\\ =5.55*10^{14} Hz$$\\$ Hence, the frequency of the radiation is $5.6 * 10^{14} Hz .$

5   The ground state energy of hydrogen atom is $- 13.6 eV$ . What are the kinetic and potential energies of the electron in this state?

Solution :

Ground state energy of hydrogen atom, $E =- 13.6 eV$$\\$ This is the total energy of a hydrogen atom. Kinetic energy is equal to the negative of the total energy. Kinetic energy $=-E =-(-13.6 )=13.6 e V$$\\$ Potential energy is equal to the negative of two times of kinetic energy. Potential energy = $-2 *( 13.6 )=- 27.2 e V$