Terms associated with elements :

• Atomic Number (Z) = No. of protons

Electron = Z – C (charge on atom)

• Mass number (A) = Total number of neutron and proton present

$\therefore$ A = Number of proton + Number of Neutrons

Electron is revolving around the nucleus in circular path.

Radius of any nucleus is

${\mathrm{R}}_{\mathrm{N}}={\mathrm{R}}_0(\mathrm{A}{)}^{1/3},{\mathrm{R}}_0=1.33\times 10^{-13}\mathrm{cm}$

$[\mathrm{A}=$ mass number, ${\mathrm{R}}_{\mathrm{N}}=$ Radius of nucleus $]$

(Radiowaves > Microwaves > Infrared rays > Visible rays > Ultraviolet rays > X-rays > Gamma Rays > Cosmic rays)

Velocity(V) = Frequency($\nu$) $\times$ Wavelength($\lambda$)

$\Rightarrow$ V = $\nu$$\lambda$

Electromagnetic waves always travels with speed of light so it's velocity(V) = C = 3 $\times$ 10⁸ m/s.

$\therefore$ C = $\nu$$\lambda$

Energy $E$ and frequency $v$ of an electromagnetic radiation are related by the Planck's equation that is

$$ E=h v $$ = $${{hc} \over \lambda }$$

$h \equiv$ Planck's constant $=6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}$

If wavelength is given in $${\mathop A\limits^ \bullet }$$ then

$\mathrm{E}=\frac{12400}{\lambda(\mathop A\limits^ \bullet)}$ eV

or $\mathrm{E}=\frac{1.984 \times {10^{ - 15}}}{\lambda(\mathop A\limits^ \bullet)}$ J

(i) Bohr model is applicable only for single e^$$-$$ system.

(ii) Out of these infinite available circular orbits electron will revolve only in those circular orbits in which its angular momentum is integer multiple of $${h \over {2\pi }}$$.

and this is known as Quantization of Angular Momentum.

Angular momentum = mvr

$$mvr = n{h \over {2\pi }}$$ .... (i)

m = mass of e^$$-$$, v = velocity of e^$$-$$, r = radius of orbit, h = Planks Constant 6.63 $$\times$$ 10^$$-$$34 J-s, n = any positive integer.

(iii) Electron will revolve in any of the circular orbit unless until force acting towards centre that is electro static force of attraction should be equal to force acting away from the centre that is centrifugal force $$\left( {{{m{v^2}} \over r}} \right)$$.

$$\therefore$$ $${{m{v^2}} \over r} = {{kz{e^2}} \over {{r^2}}}$$ .... (ii)

We know-

$$mvr = {{nh} \over {2\pi }}$$ .... (i)

$${{m{v^2}} \over r} = {{kz{e^2}} \over {{r^2}}}$$ ..... (ii)

From (i) $$\to$$ $$v = {{nh} \over {2\pi }} \times {1 \over {mr}}$$ .... (iii)

Substituting value of v in (ii)

$${m \over r} \times {\left( {{{nh} \over {2\pi }} \times {1 \over {mr}}} \right)^2} = {{kz{e^2}} \over {{r^2}}}$$

$${m \over r} \times {{{n^2}{h^2}} \over {4{\pi ^2}}} \times {1 \over {{m^2}{r^2}}} = {{kz{e^2}} \over {{r^2}}}$$

$$\therefore$$ $$r = {{{n^2}{h^2}} \over {4\pi kz{e^2}m}}$$

Substituting value of all the constants, we get

$$r = 0.529 \times {{{n^2}} \over z}\mathop A\limits^o $$

We know that,

Centrifugal force on electron moving in n^th orbit = force of attraction between nucleus and electron

$\frac{m{v}_n^2}{r}=\frac{Z{e}^2}{r^2}$ (in CGS units) ....(i)

The angular momentum of an electron is given as:

$mv{r}_n=\frac{nh}{2\pi }$ .....(ii)

From eqs. (i) and (ii), we have

$v_n\left(\frac{nh}{2\pi }\right)=Z{e}^2$

$v_n=\frac{Z}{n}\left(\frac{2\pi e^2}{h}\right)$

$v_n=\frac{Z}{n}\times 2.188\times 10^8\mathrm{cm}/\sec$

We know that,

$\frac{m{v}_n^2}{r}=\frac{Z{e}^2}{r^2}$ (in CGS units) ....(i)

$mv{r}_n=\frac{nh}{2\pi }$ .....(ii)

Let the total energy of the electron be $E_n$. It is the sum of kinetic energy and potential energy.

$\begin{array}{rl}{E}_n& =\text{ kinetic energy }+\text{ potential energy }\\ & \\ & =\frac{1}{2}m{v}^2-\frac{kZ{e}^2}{r}\end{array}$

Putting the value of $m{v}_n^2$ from eq. (i),

$E_n=\frac{kZ{e}^2}{2r}-\frac{kZ{e}^2}{r}=-\frac{kZ{e}^2}{2r}$

Putting the value of $r_n$ from eq. (iii),

$E_n=-\frac{kZ{e}^2}{2}\times \frac{4{\pi }^{2}mkZ{e}^2}{n^2{h}^2}=-\frac{2{\pi }^2{Z}^2{k}^{2}m{e}^4}{n^2{h}^2}$

Putting the values of $\pi ,k,m,e$ and $h$,

$\begin{array}{rl}{E}_n=& -\frac{2\times (3.14{)}^2\times {\left(9\times 10^9\right)}^2\times \left(9.1\times 10^{-31}\right)\times {\left(1.6\times 10^{-19}\right)}^4\times Z^2}{n^2\times {\left(6.625\times 10^{-34}\right)}^2}\\ & \\ =& -\frac{21.79\times 10^{-19}\times Z^2}{n^2}\mathrm{J}\text{ per atom }\end{array}$

$\Rightarrow$ $E_n=-13.6\times \frac{Z^2}{n^2}$ eV/atom

Kinetic energy in $n$^th shell $=\frac{13.6\times Z^2}{n^2}\mathrm{eV}$

Potential energy in $n$^th shell $=\frac{-27.2\times Z^2}{n^2}\mathrm{eV}$

Energy absorbed or Energy Released is represented by $\Delta$E.

$\Delta$E = E₂ - E₁

where E₂ = Energy of Higher Orbit = $-13.6\times \frac{Z^2}{n_2^2}$

E₁ = Energy of Lower Orbit = $-13.6\times \frac{Z^2}{n_1^2}$

$\therefore$ $\Delta$E = $-13.6\times \frac{Z^2}{n_2^2}$ - $\left(-13.6\times \frac{Z^2}{n_2^2}\right)$

= $13.6\times Z^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

Definition : Minimum energy required to eject the electron from the surface of metal is known as work function of the metal.

$\bullet$ Work function of the metal = Ionization Energy of the metal.

$\bullet$ Metal having low ionization energy are form to be suitable for photo-cell.

$\bullet$ CGM is the most suitable metal for photoelectric effect.

-Threshold frequency :- ($${\nu _0}$$)

Definition : Minimum frequency required to eject the electron from the surface of the metal is known as threshold frequency.

Threshold Wavelength ($$\lambda$$₀) :-

Definition : Maximum wavelength required to eject the electron from the surface of metal is known as Threshold wavelength.

$$\bullet$$ Both are calculated from work function of the metal.

$$\phi = h{\upsilon _0} = {{hc} \over {{\lambda _0}}}$$

$$\bullet$$ If energy of photon (E_P) is greater than work function ($$\phi$$) of the metal then remaining energy will be converted into kinetic energy of emitted photo electron.

$$KE = {E_P} - \phi $$

$$ = h\upsilon - h{\upsilon _0} = h(\upsilon - {\upsilon _0})$$

$$ = {{hc} \over \lambda } - {{hc} \over {{\lambda _0}}} = hc\left( {{1 \over \lambda } - {1 \over {{\lambda _0}}}} \right)$$

Rydberg's Equation :

$$ \begin{aligned} & \frac{1}{\lambda}=\bar{v}=\mathrm{R}_{\mathrm{H}}\left[\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right] \times \mathrm{Z}^2 \\\\ & \mathrm{R}_{\mathrm{H}} \cong 109700 \mathrm{~cm}^{-1}=\text { Rydberg constant } \end{aligned} $$

- For first line of a series $n_2=n_1+1$

- Limiting spectral line (series limit) means $\mathrm{n}_2=\infty$

- $\mathrm{H}_\alpha$ line means $\mathrm{n}_2=\mathrm{n}+1$; also known as line of longest $\lambda$, shortest $v$, least $\mathrm{E}$

- Similarly $\mathrm{H}_\beta$ line means $\mathrm{n}_2=\mathrm{n}_1+2$

- When electrons de-excite from higher energy level (n) to ground state in atomic sample, then number of spectral lines observed in the $$ \text { spectrum }=\frac{\mathrm{n}(\mathrm{n}-1)}{2} $$

- When electrons de-excite from higher energy level $\left(n_2\right)$ to lower energy level $\left(n_1\right)$ in atomic sample, then number of spectral line
observed in the spectrum $$ =\frac{\left(\mathrm{n}_2-\mathrm{n}_1\right)\left(\mathrm{n}_2-\mathrm{n}_1+1\right)}{2} $$

- No. of spectral lines in a particular series $=\mathrm{n}_2-\mathrm{n}_1$

- All material particles posses wave character as well as particle character.

- $\lambda=\frac{\mathrm{h}}{\mathrm{mv}}=\frac{\mathrm{h}}{\mathrm{p}}$

- The circumference of the $\mathrm{n}^{\text {th }}$ orbit is equal to $\mathrm{n}$ times of wavelength of electron i.e., $2 \pi \mathrm{r}_{\mathrm{n}}=\mathrm{n} \lambda$

Number of waves $=\mathrm{n}=$ principal quantum number

- Wavelength of electron $(\lambda) \cong \sqrt{\frac{150}{\mathrm{~V}(\text { volts })}}$$$\mathop A\limits^o $$

- $\lambda=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mKE}}}$

According to this principle, " it is impossible to measure simultaneously the position and momentum of a microscopic particle with absolute accuracy"

If one of them is measured with greater accuracy, the other becomes less accurate.

- $\Delta \mathrm{x} . \Delta \mathrm{p} \geq \frac{\mathrm{h}}{4 \pi}$ or $(\Delta \mathrm{x})(\Delta \mathrm{v}) \geq \frac{\mathrm{h}}{4 \pi \mathrm{m}}$

where $\Delta \mathrm{X}=$ Uncertainty in position

$\Delta \mathrm{p}=$ Uncertainty in momentum

$\Delta \mathrm{V}=$ Uncertainty in velocity

$\mathrm{m}=$ mass of microscopic particle

- Heisenberg replaced the concept of orbit by that of orbital.