Irodov Solutions: Properties of Atoms. Spectra- 3 | I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE PDF Download

Q.131. A beam of light of frequency ω, equal to the resonant frequency of transition of atoms of gas, passes through that gas heated to temperature T. In this case hω >> kT. Taking into account induced radiation, demonstrate that the absorption coefficient of the gas x varies as x = x_o  (1. — e ^-hω/kT), where x_o  is the absorption coefficient for T₀. 

Ans.  Because of the resonant nature of the processes we can ignore nonresonant processes. We also ignore spontaneous emission since it does not contribute to the absorption coefficient and is a small term if the beam is intense enough.
Suppose I is the intensity of the beam at some point. The decrease in the value of this intensity on passing through the layer of the substance of thickness d x is equal to

Here N₁ = N_o. of atoms in lower level
N₂ = N_o of atoms in the upper level per unit volume.
B₁₂ , B₂₁ are E in stein coefficients and
I_c = energy density in the beam ,
c = velocity of light

A factor  arises because each transition result in a loss or gain of energy

Hence 

But     g₁B₁₂ = g₂B₂₁ so

By Boltzman factor 

When  we can put N₁= N₀ the total number of atoms per unit volume. 

Then 

where is the absorption coefficient for T → 0.

Q.132. The wavelength of a resonant mercury line is λ = = 253.65 nm. The mean lifetime of mercury atoms in the state of resonance excitation is ζ = 0.15 μs. Evaluate the ratio of the Doppler line broadening to the natural linewidth at a gas temperature T = 300 K.

Ans. A short lived state of mean life T has an uncertainty in energy of   which is transmitted to the photon it emits as natural broadening. Then

The Doppler broadening on the other hand arises from the thermal motion of radiating atoms.
The effect is non-relativistic and the maximum broadening can be written as

Thus 

Substitution gives using  

Note: Our formula is an order of magnitude estimate.

Q.133. Find the wavelength of the K_α  line in copper (Z = 29) if the wavelength of the K_α  line in iron (Z = 26) is known to be equal to 193 pm. 

Ans. From Moseley’s law

or 

Thus  

Substitution gives

Q.134. Proceeding from Moseley's law find:
 (a) the wavelength of the K_α   line in aluminium and cobalt: 
 (b) the difference in binding energies of K and L electrons in vanadium. 

Ans. (a) From Moseley’s law

or   

We shall take σ = 1. For Aluminium  

and for cobalt  (= 27)

(b) This difference is nearly equal to the energy of the K_α line which by Moseley’s law is equal to (= 23 for vanadium)

Q.135. How many elements are there in a row between those whose wavelengths of K_α lines are equal to 250 and 179 pm?

Ans. We calculate the  values corresponding to the given wavelengths using Moseley’s law. See problem (134).
Substitution gives that

        = 23 corresponding to λ = 250 pm 

and   = 27 corresponding to λ = 179 pm

There are thus three elements in a row between those whose wavelengths of K_α lines are equal to 250 pm and 179 pm.

Q.136. Find the voltage applied to an X-ray tube with nickel anticathode if the wavelength difference between the K_α  line and the short-wave cut-off of the continuous X-ray spectrum is equal to 84 pm. 

Ans. From Moseley’s law

where  = 28 for N_i . Substitution gives

Now the short wave cut of off the continuous spectrum must be more energetic (smaller wavelength) otherwise K_α lines will not emerge. Then since  we get

λ = 82.5 pm

This corresponds to a voltage of

Substitution gives V = 15.0 kV

Q.137. At a certain voltage applied to an X-ray tube with aluminium anticathode the short-wave cut-off wavelength of the continuous X-ray spectrum is equal to 0.50 nm. Will the K series of the characteristic spectrum whose excitation potential is equal to 1.56 kV be also observed in this case? 

Ans. Since the short wavelength cut off of the continuous spectrum is

λ_o = 0.50nm

the voltage applied m ust be 

since this is greater than the excitation potential of the K series of the characteristic spectrum (which is only 1.56 k V ) the latter will be observed.

Q.138. When the voltage applied to an X-ray tube increased from V₁  =10kV to V₂ = 20 kV, the wavelength interval between the K_α  line and the short-wave cut-off of the continuous X-ray spectrum increases by a factor n = 3.0. Find the atomic number of the element of which the tube's anticathode is made. 

Ans. Suppose λ_o = wavelength of the characteristic X-ray line. Then using the formula for short wavelength limit of continuous radiation

Hence 

Using also Moseley’s law, we get

Q.139. What metal has in its absorption spectrum the difference between the frequencies of X-ray K and L absorption edges equal to Δω = 6.85.10¹⁸ s^-1 ? 

Ans. The difference in frequencies of the K and L absorption edges is equal, according to the Bohr picture, to the frequency of the K_α line (see the diagram below). Thus by Moseley’s formule

or

The metal is titanium,

Q.140. Calculate the binding energy of a K electron in vanadium whose L absorption edge has the wavelength λ_L = 2.4 nm. 

Ans. From the diagram above w e see th at the binding energy E_b of a K electron is the sum o f the energy of a K_α line and the energy corresponding to the L edge of absorption spectrum

For vanadium  = 23 and the energy of K_α line of vanadium has been calculated in problem 134 (b). Using

we get                   E_b = 5.46 k eV

Q.141. Find the binding energy of an L electron in titanium if the wavelength difference between the first line of the K series and its short-wave cut-off is Δλ = 26 pm. 

Ans. By Moseley’s law

where - E_K is the energy of the K electron and - E_L of the L electron. Also the energy of the line corresponding to the short wave cut off of the K series is

Hence 

Substitution gives for titanium ( = 22)

ω = 6.85 x 10¹⁸ s^-1

and hence E_L = 0.47keV

Q.142. Find the kinetic energy and the velocity of the photoelectrons liberated by K_α radiation of zinc from the K shell of iron whose K band absorption edge wavelength is λ_K = 174 pm. 

Ans. The energy of the K a radiation of _n is

where E = atomic number of  zinc = 30. Th e binding energy of the K electrons in iron is obtained from the wavelength of K absorption edge as.  Hence by Einstein equation

Substitution gives

T = 1.463 k eV

This corresponds to a velocity of the photo electrons of

v = 2.27x10⁶m/s

Q.143. Calculate the Lande g factor for atoms (a) in S states; (b) in singlet states. 

Ans. From the Lande formula

(a) For S states L = 0. This implies J = S. Then, if S = 0

g = 2

(For singlet states g is not defined if L = 0 )

(b) For singlet states, J = L

Q.144. Calculate the Lande g factor for the following terms: (a) ⁶F_1/2; (b) ⁴D_1/2; (c) ⁵F₂; (d) ⁵P₁; (e) ³P_o. 

Ans. (a)   Here  

(b) Here 

(c) ⁵F₂ Here S = 2 , L = 3 , J = 2

(d) ⁵P₁ Here S = 2 , L = 1 , J = 1

(e) ³P₀. For states with J = 0, L = S the g gactor is indeterminate.

Q.145. Calculate the magnetic moment of an atom (in Bohr magnetons) (a) in ¹F state; (b) in ²D_3/2  state; (c) in the state in which S = 1, L = 2, and Lande factor g = 4/3. 

Ans. (a) For the lF state S = 0, L = 3 = J

Hence 

(b) For the 

Hence 

(c) We have 

or 

or 

Hence 

Q.146. Determine the spin angular momentum of an atom in the state D₂ if the maximum value of the magnetic moment projection in that state is equal to four Bohr magnetons. 

Ans.  The expression for the projection of the magnetic moment is

where mj is the projection of 

Maximum value of the m_j is J. Thus

gJ = 4

Since J = 2 , we get g = 2. Now

Hence      S (S + 1) = 12 or S = 3

Thus 

Q.147. An atom in the state with quantum numbers L = 2, S = 1. is located in a weak magnetic field. Find its magnetic moment if the least possible angle between the angular momentum and the field direction is known to be equal to 30°. 

Ans. The angle between the angular momentum vector and the field direction is the least when the angular mommentum projection is maximum i.e. 

Thus 

or 

Hence         J = 3

Then 

and  

Q.148. A valence electron in a sodium atom is in the state with principal quantum number n = 3, with the total angular momentum being the greatest possible. What is its magnetic moment in that state? 

Ans. For a state with n = 3 , l = 2. Thus the state with maximum angular momentum is

Then 

Hence