Nuclear Physics Concept Page - 6

Example

Wavelength of different spectral lines

The minimum wavelength of paschen series of hydrogen atom will be
For Paschen Series, n=3,$n=3,$ and mimimum wavelength occurs when transition occurs just from series limit,
hCλ=−13.6[1n21−1n22]$\frac{hC}{\lambda }=-13.6[{\displaystyle \frac{1}{n_1^2}}-{\displaystyle \frac{1}{n_2^2}}]$
⇒λ=−4.14×10−15×3×10813.6×[19−1]$\Rightarrow \lambda ={\displaystyle \frac{-4.14\times {10}^{-15}\times 3\times {10}^8}{13.6\times [{\displaystyle \frac{1}{9}}-1]}}$
⇒λ=4.14×10−15×3×108×913.6×8$\Rightarrow \lambda ={\displaystyle \frac{4.14\times {10}^{-15}\times 3\times {10}^8\times 9}{13.6\times 8}}$
⇒λ=1.878655×10−6×9×0.04861111$\Rightarrow \lambda =1.878655\times {10}^{-6}\times 9\times 0.04861111$ m
⇒λ=8.21911×10−07$\Rightarrow \lambda =8.21911\times {10}^{-07}$m
⇒λ=8219.11×10−10$\Rightarrow \lambda =8219.11\times {10}^{-10}$ m
⇒λ=8219A∘≈8181A∘$\Rightarrow \lambda =8219A^{\circ }\approx 8181A^{\circ }$

Definition

Wavelength of a spectral line emitted by mercury and sodium lamps

The emission spectrum of atomic hydrogen is divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in the atom. 
Rydberg formula:1λ=RZ2(1n′2−1n2)${\displaystyle \frac{1}{\lambda }}=R{Z}^2({\displaystyle \frac{1}{{n^{\prime }}^2}}-{\displaystyle \frac{1}{n^2}})$ where Z=1 for hydrogen atom.
Now for sodium and mercury put Z=11$Z=11$ and Z=80$Z=80$ respectively and calculate the wavelength of the light used.  

Definition

Rydberg constant

The Rydberg constant represents the limiting value of the highest  wavenumber (the inverse wavelength) of any photon that can be emitted from the hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing the hydrogen atom from its ground state.
R=me48ϵ20h3c=1.0973×107m−1$R={\displaystyle \frac{m{e}^4}{8{ϵ}_0^2{h}^{3}c}}=1.0973\times {10}^7{m}^{-1}$

Definition

Wavelength of different spectral series

Lyman series:
1λ=R(112−1n2)${\displaystyle \frac{1}{\lambda }}=R({\displaystyle \frac{1}{1^2}}-{\displaystyle \frac{1}{n^2}})$  n=2,3,4...
Balmer series:
1λ=R(122−1n2)${\displaystyle \frac{1}{\lambda }}=R({\displaystyle \frac{1}{2^2}}-{\displaystyle \frac{1}{n^2}})$  n=3,4,5...
Paschen series:
1λ=R(132−1n2)${\displaystyle \frac{1}{\lambda }}=R({\displaystyle \frac{1}{3^2}}-{\displaystyle \frac{1}{n^2}})$  n=4,5,6...
Brackett series:
1λ=R(142−1n2)${\displaystyle \frac{1}{\lambda }}=R({\displaystyle \frac{1}{4^2}}-{\displaystyle \frac{1}{n^2}})$  n=5,6,7...
Pfund series:
1λ=R(152−1n2)${\displaystyle \frac{1}{\lambda }}=R({\displaystyle \frac{1}{5^2}}-{\displaystyle \frac{1}{n^2}})$  n=6,7,8

Definition

De Broglie's Justification of Bohr's Assumption

De Broglie came up with an explanation for why the angular momentum might be quantized in the manner Bohr assumed it was. De Broglie realized that if you use the wavelength associated with the electron, and assume that an integral number of wavelengths must fit in the circumference of an orbit, you get the same quantized angular momenta that Bohr did.
The circumference of the circular orbit must be an integral number of  wavelengths:
2πr=nλ=nhp$2\pi r=n\lambda ={\displaystyle \frac{nh}{p}}$          (λ=hp)$(\lambda ={\displaystyle \frac{h}{p}})$
The momentum, p, is simply mv as long as we're talking about non-relativistic speeds, so this becomes:
2πr=nhmv$2\pi r={\displaystyle \frac{nh}{mv}}$
Rearranging this a little gives the Bohr relationship:
Lr=mvr=nh2π$L_r=mvr={\displaystyle \frac{nh}{2\pi }}$

Definition

Isotopes

The atoms belonging to the same element, having same atomic number Z, but different mass number A, are called isotopes. For example, carbon-12, carbon-13 and carbon-14 are three isotopes of the element carbon with mass numbers 12, 13 and 14 respectively.

Example

Relative Abundance of an isotope

Relative abundance is the percent composition of an isotope of a particular kind relative to the total number of isotopes. Relative species abundances tend to conform to specific patterns that are among the best-known and most-studied patterns.

Definition

Isobars

The atoms of different elements which have the same mass number A, but different atomic number Z, are called isobars. An example of a series of isobars would be 40S,40Cl,40Ar,40K${}^{40}S{,}^{40}Cl{,}^{40}Ar{,}^{40}K$, and 40Ca${}^{40}Ca$. The nuclei of these nuclides all contain 40 nucleons; however, they contain varying numbers of protons and neutrons.

Definition

Isotones

The atoms having different number of protons but same number of neutrons i.e., different Z and A, but same A - Z are called isotones. They have different number of electrons. For example, boron-12 and carbon-13 nuclei both contain 7 neutrons, and so are isotones .

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