Electricity and Magnetism - Nuclear atom

A neutral atom of Carbon-14 is represented as ${}_{6}^{14}\text{C}$. Determine the number of protons, neutrons, and electrons in this atom.

Protons: $6$, Neutrons: $8$, Electrons: $6$.

The proton number $Z$ is the subscript ($6$). Since it is a neutral atom, the number of electrons equals the number of protons ($6$). The neutron number $N$ is calculated by $A - Z$, which is $14 - 6 = 8$.

Calculate the total electric charge of a Gold nucleus (${}_{79}^{197}\text{Au}$), given the elementary charge $e = 1.6 \times 10^{-19} \text{ C}$.

$Q = +1.264 \times 10^{-17} \text{ C}$

The nucleus contains only protons and neutrons. The charge is determined by the number of protons ($Z = 79$). Using $Q = Ze$, we get $Q = 79 \times (1.6 \times 10^{-19} \text{ C}) = 1.264 \times 10^{-17} \text{ C}$.

Uranium-238 (${}_{92}^{238}\text{U}$) undergoes alpha decay to become Thorium (Th). Write the balanced nuclear equation.

${}_{92}^{238}\text{U} \rightarrow {}_{90}^{234}\text{Th} + {}_{2}^{4}\alpha$

In $\alpha$-decay, the nucleus loses $2$ protons and $2$ neutrons. Therefore, the nucleon number $A$ decreases by $4$ ($238 - 4 = 234$) and the proton number $Z$ decreases by $2$ ($92 - 2 = 90$).