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Physics Part II

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Physics

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In a hydrogen like atom, electron makes the transition from an energy level with quantum number n to another with a quantum number (n – 1). If n >> 1, the frequency of radiation emitted is proportional to 

  • 1/n

  • 1/n2

  • 1/n3/2

  • 1/n3


D.

1/n3

An energy gap, ΔE = hv
Here, h is Planck's constant
therefore,
Frequency
straight v space equals space fraction numerator increment straight E over denominator straight h end fraction space equals space straight k space open square brackets fraction numerator 1 over denominator left parenthesis straight n minus 1 right parenthesis squared end fraction minus 1 over straight n squared close square brackets
rightwards double arrow space straight v space equals fraction numerator straight k 2 straight n over denominator straight n squared left parenthesis straight n minus 1 right parenthesis squared end fraction space
straight v space proportional to 1 over straight n cubed

An energy gap, ΔE = hv
Here, h is Planck's constant
therefore,
Frequency
straight v space equals space fraction numerator increment straight E over denominator straight h end fraction space equals space straight k space open square brackets fraction numerator 1 over denominator left parenthesis straight n minus 1 right parenthesis squared end fraction minus 1 over straight n squared close square brackets
rightwards double arrow space straight v space equals fraction numerator straight k 2 straight n over denominator straight n squared left parenthesis straight n minus 1 right parenthesis squared end fraction space
straight v space proportional to 1 over straight n cubed

398 Views

The radiation corresponding to 3 → 2 transition of hydrogen atom falls on a metal surface to produce photoelectrons. These electrons are made to enter a magnetic field of 3 x 10-4 T. If the radius of the largest the circular path followed by these electrons is 10.0 mm, the work function of the metal is close to:

  • 1.8 eV

  • 1.1 eV

  • 0.8 eV

  • 1.6 eV


B.

1.1 eV

mv =qBR

KE subscript max space equals space fraction numerator open parentheses mv close parentheses squared over denominator 2 straight m end fraction space equals space 0. space 8 space eV
hv space equals space 13.6 space open square brackets 1 fourth minus 1 over 6 close square brackets
therefore space straight w space equals space hv minus KE subscript max
space equals space 13.6 5 over 36 minus 0.8 space equals space 1.1 space eV

mv =qBR

KE subscript max space equals space fraction numerator open parentheses mv close parentheses squared over denominator 2 straight m end fraction space equals space 0. space 8 space eV
hv space equals space 13.6 space open square brackets 1 fourth minus 1 over 6 close square brackets
therefore space straight w space equals space hv minus KE subscript max
space equals space 13.6 5 over 36 minus 0.8 space equals space 1.1 space eV

651 Views

Hydrogen (1H1), deuterium (1H2), singly ionised helium (2He4+) and doubly ionised lithium (3Li8)2+ all have one electron around the nucleus. Consider an electron transition from n =2 to n=1. If the wavelengths of emitted radiation are λ123 andλ4, respectively for four elements, then approximately which one of the following is correct?

  •  4λ1=2λ2=2λ3 =λ4

  •  λ1=2λ2=2λ3 =λ4

  •  λ12=4λ3 =9λ4

  •  λ1=2λ2=3λ3 =4λ4


C.

 λ12=4λ3 =9λ4

For hydrogen atom, we get

1 over straight lambda space equals space Rz squared open parentheses 1 over 1 squared minus 1 over 2 squared close parentheses
1 over straight lambda subscript 1 space equals space straight R space left parenthesis 1 right parenthesis squared open parentheses 3 over 4 close parentheses
fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 2 end style end fraction space equals space straight R space left parenthesis 1 right parenthesis squared open parentheses fraction numerator begin display style 3 end style over denominator begin display style 4 end style end fraction close parentheses
fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 3 end style end fraction space equals space straight R space left parenthesis 2 right parenthesis squared open parentheses fraction numerator begin display style 3 end style over denominator begin display style 4 end style end fraction close parentheses
fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 4 end style end fraction space equals space straight R space left parenthesis 3 right parenthesis squared open parentheses fraction numerator begin display style 3 end style over denominator begin display style 4 end style end fraction close parentheses
fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 1 end style end fraction space equals space fraction numerator begin display style 1 end style over denominator begin display style 4 straight lambda subscript 3 end style end fraction equals fraction numerator begin display style 1 end style over denominator begin display style 9 straight lambda subscript 4 end style end fraction space equals fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 2 end style end fraction

For hydrogen atom, we get

1 over straight lambda space equals space Rz squared open parentheses 1 over 1 squared minus 1 over 2 squared close parentheses
1 over straight lambda subscript 1 space equals space straight R space left parenthesis 1 right parenthesis squared open parentheses 3 over 4 close parentheses
fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 2 end style end fraction space equals space straight R space left parenthesis 1 right parenthesis squared open parentheses fraction numerator begin display style 3 end style over denominator begin display style 4 end style end fraction close parentheses
fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 3 end style end fraction space equals space straight R space left parenthesis 2 right parenthesis squared open parentheses fraction numerator begin display style 3 end style over denominator begin display style 4 end style end fraction close parentheses
fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 4 end style end fraction space equals space straight R space left parenthesis 3 right parenthesis squared open parentheses fraction numerator begin display style 3 end style over denominator begin display style 4 end style end fraction close parentheses
fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 1 end style end fraction space equals space fraction numerator begin display style 1 end style over denominator begin display style 4 straight lambda subscript 3 end style end fraction equals fraction numerator begin display style 1 end style over denominator begin display style 9 straight lambda subscript 4 end style end fraction space equals fraction numerator begin display style 1 end style over denominator begin display style straight lambda subscript 2 end style end fraction

326 Views

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 ni ) to the lower state, (nf ).

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


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 En,

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 rn 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 ni and nf are the quantum numbers of initial and final states and Ei & Ef 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 ni = 4 to nf = 3.21 .

Radiation belongs to Paschen, Balmer and Lyman series.

4148 Views

As an electron makes a transition from an excited state to the ground state of a hydrogen – like atom/ion:

  • kinetic energy, potential energy and total energy decrease

  • kinetic energy decreases, potential energy increases but total energy remains same

  • kinetic energy and total energy decrease but potential energy increases

  • its kinetic energy increases but the potential energy and total energy decrease


A.

kinetic energy, potential energy and total energy decrease

As we know that kinetic energy of an electron is 
KE ∝ (Z/n)

when the electron makes the transition from an excited state of the ground state then n, decreases and KE increases. We know that PE is lowest for the ground state. As TE=- KE and TE also decreases.

As we know that kinetic energy of an electron is 
KE ∝ (Z/n)

when the electron makes the transition from an excited state of the ground state then n, decreases and KE increases. We know that PE is lowest for the ground state. As TE=- KE and TE also decreases.

284 Views