Dinitrogen oxide, \(\mathrm{N}_{2} \mathrm{O}\), gas was generated from the thermal decomposition of ammonium nitrate and collected over water. The wet gas occupied \(126 \mathrm{~mL}\) at \(21^{\circ} \mathrm{C}\) when the atmospheric pressure was 755 Torr. What volume would the same amount of \(d r y\) dinitrogen oxide have occupied if collected at 755 Torr and \(21^{\circ} \mathrm{C}\) ? The vapor pressure of water is \(18.65\) Torr at \(21^{\circ} \mathrm{C}\).

Boyle's Law is a fundamental principle in understanding how gases behave at constant temperatures. Named after the chemist Robert Boyle, this law states that the pressure of a gas is inversely proportional to its volume, as long as the temperature and the amount of gas remain constant.

In equation form, Boyle's Law is expressed as: \[ P_1 V_1 = P_2 V_2 \] where \( P_1 \) and \( V_1 \) are the initial pressure and volume respectively, and \( P_2 \) and \( V_2 \) are the final pressure and volume. This law is incredibly useful when we need to predict how changing the pressure of a gas will affect its volume, or vice versa, without a change in temperature or the amount of gas.

For instance, if you squeeze a gas into a smaller space, its pressure increases as the volume decreases. Conversely, if you allow a gas to expand, its volume increases and the pressure decreases. This concept plays a crucial role in various scientific and industrial processes where the control of gas volumes and pressures is necessary.

The Ideal Gas Law bridges together Boyle’s Law with other gas laws, providing a comprehensive equation that models the behavior of an ideal gas under various conditions. It is described by the formula: \[ PV = nRT \] where \( P \) is the pressure of the gas, \( V \) is the volume, \( n \) is the number of moles of the gas, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

The Ideal Gas Law is powerful because it allows us to predict how a gas will behave when subjected to different pressures, volumes, temperatures, or changes in the amount of the gas present. In classroom settings, it’s often used to solve problems involving dry gases - gases not influenced by the presence of a liquid. When the constants \( n \) and \( R \), and one of the variables \( P \), \( V \), or \( T \) are known, the other variables can be calculated easily.

In the context of our textbook solution, although the Ideal Gas Law encompasses Boyle's Law within it, for the exercise considered, where the temperature \( (T) \) is constant and we’re dealing with a constant number of moles of gas, Boyle’s Law alone is sufficient to determine the change in volume of the gas.

Thermal decomposition is a chemical reaction where a compound breaks down into two or more substances when heated. It is an essential concept in the study of chemical reactions and is commonly encountered in the production of gases in a laboratory.

During thermal decomposition, the reactant is subjected to high temperatures until it reaches a point where it can no longer maintain its chemical structure, leading to its breakdown. For example, when ammonium nitrate is heated, it decomposes into dinitrogen oxide (also known as nitrous oxide or laughing gas) and water vapor. This process can be represented by the following chemical equation: \[ NH_4NO_3 \rightarrow N_2O + 2H_2O(g) \]

The gases produced in thermal decomposition may include not only the desired product but also by-products like water vapor as observed in our exercise. The presence of such by-products can alter the total pressure of the gas mixture collected, which is a primary consideration for applying Dalton's Law of Partial Pressures when we are isolating the volume of the desired dry gas.