Write the balanced nuclear equation for each of the following decays: (a) \(\beta^{-}\)decay of uranium-233; (b) proton emission of cobalt- 56 ; (c) \(\beta^{+}\)decay of holmium-158; (d) \(\alpha\) decay of polonium-212.

Beta decay is a nuclear process where a radioactive atom emits a beta particle and transforms into another element. During \textbf{\beta^{-} decay}, a neutron in the nucleus is converted into a proton, an electron (the beta particle), and an antineutrino. It's important to remember that in beta decay, the mass number remains the same because a neutron is simply converted into a proton; however, the atomic number increases by one, indicating a new element has formed.

For example, when uranium-233 undergoes beta decay, the balanced nuclear equation is: \[ ^{233}_{92}U \rightarrow ^{233}_{93}Np + ^{-1}_{0}\beta + \bar{u} \] Where the subscript and superscript on the beta particle \( ^{-1}_{0}\beta \) show the change in atomic and mass numbers, respectively, and \( \bar{u} \) symbolizes the antineutrino.

Proton emission is a less common type of nuclear decay that involves the ejection of a proton from the nucleus. This decay mode can occur in very proton-rich nuclei. When a nucleus emits a proton, both the atomic and the mass numbers decrease by one. The daughter nucleus then has one less proton and, as such, becomes a different element.

To illustrate, the decay of cobalt-56 through proton emission can be represented by the equation: \[ ^{56}_{27}Co \rightarrow ^{55}_{26}Fe + ^{1}_{1}p \] This equation shows cobalt-56 transforming into iron-55 and releasing a proton \( ^{1}_{1}p \).

Similarly to beta decay, positron emission or \textbf{\beta^{+} decay}, involves the transformation of a proton into a neutron within the atomic nucleus, but instead of emitting an electron, it emits a positron (the antimatter counterpart of the electron) and a neutrino. The result is a decrease in the atomic number by one, while the mass number stays unchanged.

A prime example is the decay of holmium-158, described by the equation: \[ ^{158}_{67}Ho \rightarrow ^{158}_{66}Dy + ^{0}_{1}\beta^{+} + u \] Here, \( ^{0}_{1}\beta^{+} \) is the emitted positron and \( u \) represents the neutrino, showing that holmium-158 decays into dysprosium-158 with the emission of a positron.

Alpha decay is another fundamental type of radioactive decay in which an unstable nucleus emits an alpha particle, consisting of two protons and two neutrons (identical to a helium-4 nucleus). This process results in a decrease of the atomic number by two and the mass number by four. It's a common decay method observed in heavy elements such as uranium and thorium.

A clear example of this is seen in the alpha decay of polonium-212: \[ ^{212}_{84}Po \rightarrow ^{208}_{82}Pb + ^{4}_{2}\text{He} \] or simply \[ ^{212}_{84}Po \rightarrow ^{208}_{82}Pb + ^{4}_{2}\alpha \] The resulting element is lead-208, and the emitted \( ^{4}_{2}\alpha \) particle is identical to a helium nucleus.

Balancing nuclear reactions requires careful attention to the conservation laws governing nuclear physics: the conservation of mass number and the conservation of atomic number. Every nuclear equation must be balanced in such a way that the sum of mass numbers (top number) and atomic numbers (bottom number) on the reactant side equals those on the product side. This principle ensures that the identity and the total number of protons and neutrons remain the same before and after the reaction.

For each type of decay previously discussed, balancing the equation entailed making sure the sum of the mass numbers and the sum of the atomic numbers were equal on both sides of the equation. This fundamental technique verifies the accuracy of the nuclear transformation being examined.