### The Nucleus

- existence and size demonstrated using the Rutherford $$\alpha$$-scattering experiment.
- consists of
**nucleons**(protons and neutrons) - isotopes of an element share the same number of protons but different number of neutrons.

### Nuclear Reactions

- nuclear reactions involve two or more reactants.
- represented using the form: $${^{14}_7N}+{^4_2He}\rightarrow{^{17}_8O}+{^1_1H}$$
- for a reaction that releases energy, mass-energy of reactants = mass-energy of products +
*E*,

where $$E = mc^2$$ and*m*is the**mass defect**(difference in mass between the products and reactants). **binding energy**is the energy released when the nucleus is formed from its separate protons and neutrons. The same amount of energy is required to break up a nucleus into its constituent nucleons.

**binding energy per nucleon**($$\frac{B.E.}{A}$$) is an indication of the stability of a nucleus, where*B.E*.is binding energy and*A*is the nucleon number. You need to know how to sketch its variation with nucleon number. (The following video explains the shape of the $$\frac{B.E.}{A}$$ versus*A*graph and why it peaks at $$^{56}Fe$$.

**nuclear fission**is the disintegration of a heavy nucleus into two lighter nuclei of comparable mass with the emission of neutrons and release of energy.

e.g. $${^{235}_{92}U}+{^1_0n}\rightarrow{^{236}_{92}U}\rightarrow{^{144}_{56}Ba}+{^{90}_{36}Kr}+2^1_0n+Energy$$**nuclear fusion**occurs when two light nuclei combine to form a single more massive nucleus, leading to the release of energy.

e.g. $${^2_1H}+{^3_1H}\rightarrow{^4_2He}+{^1_0n}+Energy$$

- The following quantities are always conserved:
- proton number & neutron number
- momentum
- mass-energy

### Radioactive Decay

- spontaneous and random emission of radiation from a radioactive nucleus.
**$$\alpha$$ particle**– helium nucleus**$$\beta$$ particle**– electron**$$\gamma$$ particle**– electromagnetic radiation

- $$A=-\frac{dN}{dt}=\lambda N$$

where*A*is the rate of disintegration or**activity**,*N*is the number of radioactive nuclei and $$\lambda$$ is the**decay constant**. - $$x=x_0{e^{-\lambda t}}$$

where*x*could represent the activity, number of undecayed particles or received count rate. **half-life**($$t_{\frac{1}{2}}$$) is the average time taken for half the original number of radioactive nuclei to decay.- From $$x=x_0{e^{-\lambda t}}$$,

$$\frac{x}{x_0}=\frac{1}{2}=e^{-\lambda t_{\frac{1}{2}}}$$

$$\Rightarrow{-ln2}=-\lambda t_{\frac{1}{2}}$$

$$\Rightarrow{t_{\frac{1}{2}}}=\frac{ln 2}{\lambda}$$ - You may also use $${\frac{x}{x_0}}={\frac{1}{2}}^{\frac{t}{t_{1/2}}}$$, as shown in the following video.