Modem Physics Synopsis
Photon’s energy, \(E=h f=\frac{h c}{\lambda}=\frac{1240 \mathrm{eV} \mathrm{nm}}{\lambda}\)
Photon’s momentum, \(p=\frac{E}{c}=\frac{h}{\lambda}\).
Einstein’s photoelectric equation:
hf = Φ + KEmax,
where f = frequency of the incident light
and Φ = photoelectric work function = \(h f_0=\frac{h c}{\lambda_0}\)
For photoemissions, hf ≥ Φ ⇒ \(\lambda \leq \lambda_0=\frac{h c}{\phi}\)
Stopping potential (Vs): KEmax = eVs => hf = Φ0 + eVs.
The de Broglie wavelength of a matter wave is given by
⇒ \(\lambda_{\mathrm{dB}}=\frac{h}{p}=\frac{h}{m v}\)
∵ \(\mathrm{KE}=\frac{p^2}{2 m} \Rightarrow p=\sqrt{2 m E_{\mathrm{k}}}\),
∴ \(\lambda_{\mathrm{dB}}=\frac{h}{\sqrt{2 m E_{\mathrm{k}}}}\)
The wavelength of an electron accelerated by a voltage V: This is given by
∴ \(\lambda=\frac{h}{\sqrt{2 m e V}}=\frac{1.227}{\sqrt{V}} \mathrm{~nm}=\sqrt{\frac{150}{V}}\) Å.
Bohr’s quantum conditions:
Angular momentum, L = mvrn = \(n\left(\frac{h}{2 \pi}\right)\)
Radius of the nth orbit, rn = \(\left(\frac{a_0}{Z}\right)\) n2,
where a0 = Bohr radius = 0.53 Å.
Speed of an electron in the nth orbit, \(v_n=\frac{Z e^2}{2 \varepsilon_0 h} \cdot \frac{1}{n}\)
So, \(v_n \propto \frac{1}{n}\).
KE in the nth orbit, \(\mathrm{KE}_n=\frac{Z e^2}{8 \pi \varepsilon_0 r_n}=Z^2\left(\frac{13.6}{n^2}\right) \mathrm{eV}\).
PE in the nth orbit, \(\mathrm{PE}_n=-\frac{Z e^2}{4 \pi \varepsilon_0 r_n}=-2 \mathrm{KE}_n\).
The total energy in the nth orbit, En = KEn + PEn
⇒ \(-\frac{\mathrm{Z} e^2}{8 \pi \varepsilon_0 r_n}\)
⇒ \(-\left(\frac{13.6}{n^2}\right) Z^2 \mathrm{eV}\)
Note that | PE| = 2KE and| Etot| = KE.
Photon energy during emission, hv = E2-E1.
Photon energy during absorption, hv = E2-E1.
The energy of the emitted photons, ΔE = Ei-Ef
⇒ \(h v=13.6 \mathrm{Z}^2\left(\frac{1}{n_{\mathrm{f}}^2}-\frac{1}{n_{\mathrm{i}}^2}\right)\)
Wave number, \(\frac{1}{\lambda}=\frac{13.6 \mathrm{Z}^2}{h c}\left(\frac{1}{n_{\mathrm{f}}^2}-\frac{1}{n_{\mathrm{i}}^2}\right)=R_{\infty} Z^2\left(\frac{1}{n_{\mathrm{f}}^2}-\frac{1}{n_{\mathrm{i}}^2}\right)\),
where R∞ = Rydberg constant = 1.0973 x 107 m-1
and R∞hc = 13.6 eV.
Ionization energy = energy required to detach an electron.
∴ \(E_{\text {ion }}=\frac{13.6}{n^2} \mathrm{eV}\)
Ionization potential = \(\frac{13.6}{n^2} \mathrm{~V}\).
Maximum number of possible transitions from the nth state to the ground state, \(N=(n-1)+(n-2)+\ldots+1=\frac{n(n-1)}{2}\).
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Constitution of a nucleus: Any nucleus AXZ has the following:
-
- Z = number of protons = atomic number
- A-Z = number of neutrons
- A- mass number
Isotopes: Nuclei having the same number of protons (Z) but different mass numbers (A) are called isotopes (Example., \({ }_1^2 \mathrm{H},{ }_1^3 \mathrm{H} ;{ }_8^{17} \mathrm{O}, { }_8^{18} \mathrm{O} ;{ }_{92}^{235} \mathrm{U},{ }_{92}^{238} \mathrm{U}\)).
Isobars: Nuclei having the same mass number (A) but different numbers of protons (Z) are called isobars (Example., \({ }_1^3 \mathrm{H},{ }_2^3 \mathrm{He} ;{ }_3^7 \mathrm{Li}, { }_4^7 \mathrm{Be} ;{ }_{18}^{40} \mathrm{Ar},{ }_{20}^{40} \mathrm{Ca}\)).
Isotones: Nuclei having the same number of neutrons (A- Z) called isotones (Example., \({ }_1^3 \mathrm{H},{ }_2^4 \mathrm{He} ;{ }_7^{17} \mathrm{~N},{ }_8^{18} \mathrm{O},{ }_9^{19} \mathrm{~F}\)).
Nuclear radius, R = R0A1/3, where R0 = 1.1 x 10-15m and A = mass number.
The density of nuclear matter is independent of the mass number A, and its value is around 2.3 x 1017 kg m-3.
Atomic mass unit (in short, amu) (symbol: u)
= \(\frac{1}{12} \text { (mass of a }{ }_6^{12} \mathrm{C} \text { atom) }\).
Thus, \(1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg} \approx \frac{931.5 \mathrm{MeV}}{c^2}\).
The mass defect is the amount ΔM by which the mass of an atomic nucleus differs from the sum of the masses of its constituent particles. Thus,
ΔM = Zmp+(A-Z)mn-M.
Binding energy = ΔMc2.
Q-value of a nuclear process: Q = Ui -Uf = (mi– mf )c2 = Amc2.
Rate of disintegration (decay), \(\frac{d N}{d t}=-\lambda N\).
An instantaneous number of active nuclei, N = N0e-λt.
Activity: \(A=\left|\frac{d N}{d t}\right|=\lambda N \text { and } A(t)=A_0 \mathrm{e}^{-\lambda t}\)
Half-life, \(T_{1 / 2}=\frac{1}{\lambda} \ln 2=\frac{0.693}{\lambda}\)
Average life, \(T_{\mathrm{av}}=\frac{1}{\lambda}\)
Number of active (undecayed) nuclei after n half-lives, \(N=N_0\left(\frac{1}{2}\right)^n\).