Question

Using Bohr’s postulates, derive the expression for the frequency of radiation emitted when electron in hydrogen atom undergoes transition from higher energy state (quantum number n_i) to the lower state, (n_f ).

When electron in hydrogen atom jumps from energy state n_i =4 to n_f =3, 2, 1, identify the spectral series to which the emission lines belong.

Solution

According to Bohr’s postulates, in a hydrogen atom, a single electron revolves around a nucleus of charge +e. For an electron moving with a uniform speed in a circular orbit on a given radius, the centripetal force is provided by the Coulomb force of attraction between the electron and the nucleus.

Therefore,
$fraction numerator m v squared over denominator r end fraction space equals space fraction numerator 1 space left parenthesis Z e right parenthesis space left parenthesis e right parenthesis over denominator 4 pi epsilon subscript o r squared end fraction$ ... (1)
$rightwards double arrow m v squared space equals fraction numerator 1 space over denominator 4 pi epsilon subscript o end fraction fraction numerator Z e squared over denominator r end fraction$
So, Kinetic Energy, K.E = $1 half m v squared$
$K. E equals fraction numerator 1 over denominator 4 pi epsilon subscript o end fraction fraction numerator Z e squared over denominator r end fraction$
Potential energy is given by, P.E = $fraction numerator 1 over denominator 4 pi epsilon subscript o end fraction fraction numerator left parenthesis Z e right parenthesis space left parenthesis negative e right parenthesis over denominator r end fraction space equals space minus fraction numerator 1 space over denominator 4 pi epsilon subscript o end fraction fraction numerator Z e squared over denominator r end fraction$
Therefore, total energy is given by, E = K.E + P.E = $Error converting from MathML to accessible text.$
E = $negative fraction numerator 1 space over denominator 4 pi epsilon subscript o end fraction fraction numerator Z e squared over denominator 2 r end fraction$ , is the total energy.
For nth orbit, E can be written as E_n,
$straight E subscript straight n space equals negative space fraction numerator 1 over denominator 4 πε subscript straight o end fraction fraction numerator Ze squared over denominator 2 straight r subscript straight n end fraction italic space$ ... (2)
Now, using Bohr's postulate for quantization of angular momentum, we have
$mvr space equals space fraction numerator nh over denominator 2 straight pi end fraction$
$rightwards double arrow straight v space equals space fraction numerator nh over denominator 2 πmr end fraction$

Putting this value of v in equation (1), we get
$Error converting from MathML to accessible text.$
$rightwards double arrow space straight r space equals space fraction numerator straight epsilon subscript straight o straight h squared straight n squared over denominator πmZe squared end fraction$

$rightwards double arrow space straight r space equals space fraction numerator straight epsilon subscript straight o straight h squared straight n squared over denominator πmZe squared end fraction$

Now, putting value of r_n in equation (2), we get
$Error converting from MathML to accessible text.$
R is the rydberg constant.

For hydrogen atom Z =1,
$straight E subscript straight n space equals space fraction numerator negative Rch over denominator straight n squared end fraction$
If n_i and n_fare the quantum numbers of initial and final states and E_i & E_f are energies of electron in H-atom in initial and final state, we have

$straight E subscript straight i space equals negative space Rhc over straight n subscript straight i squared space and space straight E subscript straight f space equals space fraction numerator negative Rhc over denominator straight n subscript straight f squared end fraction$
$If comma space straight upsilon space is space the space frequency space of space emitted space radiation comma space we space get space$
$straight nu space equals space fraction numerator straight E subscript straight i space minus space straight E subscript space straight f end subscript over denominator straight h end fraction straight nu space equals space fraction numerator negative Rc over denominator straight n subscript straight i squared end fraction minus open parentheses fraction numerator negative Rc over denominator straight n subscript straight f squared end fraction close parentheses space equals space Rc open square brackets 1 over straight n subscript straight f squared space minus space 1 over straight n subscript straight i squared close square brackets$

That is, when electron jumps from n_i = 4 to n_f = 3.21 .
Radiation belongs to Paschen, Balmer and Lyman series.

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Wave Optics

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In the ground state of electrons are in stable equilibrium, while in electrons always experience a net force. (Thomson's model/Rutherford's model).

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The positively charged part of the atom possesses most of the mass in ___________ (Rutherford's model/both the models).

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?

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