In the presence of nitric acid, \(\mathrm{UO}^{2+}\) undergoes a redox process. It is converted to \(\mathrm{UO}_{2}{ }^{2+}\) and nitric oxide (NO) gas is produced according to the following unbalanced equation: \(\mathrm{H}^{+}(a q)+\mathrm{NO}_{3}^{-}(a q)+\mathrm{UO}^{2+}(a q) \longrightarrow\) \(\mathrm{NO}(g)+\mathrm{UO}_{2}^{2+}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\) If \(2.55 \times 10^{2} \mathrm{~mL} \mathrm{NO}(g)\) is isolated at \(29^{\circ} \mathrm{C}\) and \(1.5 \mathrm{~atm}\), what amount (moles) of \(\mathrm{UO}^{2+}\) was used in the reaction? (Hint: Balance the reaction by the oxidation states method.)

Redox reactions are fundamental types of chemical reactions where oxidation and reduction processes occur simultaneously. To analyze redox reactions, it is essential to understand oxidation states, as they help indicate the degree of oxidation or reduction of an atom within a compound. In a redox reaction, the substance that loses electrons is oxidized (increase in oxidation state), while the substance that gains electrons is reduced (decrease in oxidation state).

Oxidation states can be determined by following certain rules based on electronegativity, elemental states, and known charges on ions. For instance, in our exercise, the oxidation state of uranium changes from +4 to +6, which means it undergoes oxidation by losing electrons. Conversely, nitrogen goes from +5 to +2, indicating it accepts electrons and is reduced. The importance of correctly identifying oxidation states cannot be overstated, as it is the first step in balancing a redox equation.

Stoichiometry is the section of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It is based on the conservation of mass and the concept of moles, which allows chemists to calculate the amounts of substances consumed and produced. When you have a balanced chemical equation, stoichiometry becomes a powerful tool to predict how much of each reactant is needed or how much of a product will be formed.

In the provided exercise, stoichiometry is used to connect the moles of nitric oxide gas produced to the original moles of \(\mathrm{UO}^{2+}\) reacted. Given that the balanced equation shows a 3:2 ratio between \(\mathrm{UO}^{2+}\) and NO, stoichiometry allows us to establish a direct proportional relationship. By knowing the amount of one substance, we can calculate the amount of another. This makes stoichiometry a key concept for solving many practical problems in chemistry, not just in the classroom but also in industrial applications like pharmaceuticals and materials science.

The ideal gas law is a crucial equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas through the formula \(PV = nRT\). It is based on the ideal gas assumption, which treats gases as a collection of particles that have no volume and do not interact, allowing for straightforward calculations. While real gases may deviate from this ideal behavior, the ideal gas law is incredibly useful for estimating the behavior of gases under many conditions.

When applying the ideal gas law, as seen in this exercise, it's important to ensure all units are consistent, typically using liters (L) for volume, atmospheres (atm) for pressure, Kelvin (K) for temperature, and moles (mol) for the amount of substance. The gas constant \(R\) has different values depending on the units used; for the ideal gas law using atm, L, and K, \(R\) equals 0.0821 \((L\cdot atm)/(mol\cdot K)\). By knowing any three of the variables, we can solve for the fourth. This forms the basis for solving a variety of problems, including finding the amount of a gas produced or consumed in a chemical reaction, as demonstrated in the exercise.