Calculate each of the following quantities for an ideal gas: (a) the volume of the gas, in liters, if \(1.50 \mathrm{~mol}\) has a pressure of 1.25 atm at a temperature of \(-6^{\circ} \mathrm{C} ;(\mathbf{b})\) the absolute temperature of the gas at which \(3.33 \times 10^{-3}\) mol occupies \(478 \mathrm{~mL}\) at 750 torr; \((\mathbf{c})\) the pressure, in atmospheres, if \(0.00245 \mathrm{~mol}\) occupies \(413 \mathrm{~mL}\) at \(138{ }^{\circ} \mathrm{C} ;(\mathbf{d})\) the quantity of gas, in moles, if \(126.5 \mathrm{~L}\) at \(54^{\circ} \mathrm{C}\) has a pressure of \(11.25 \mathrm{kPa}\).

The molar volume of a substance is the volume that one mole of gas occupies under certain conditions of temperature and pressure. It is particularly significant when discussing ideal gases. According to the Ideal Gas Law, all ideal gases have the same molar volume at a given temperature and pressure, making it a universal property.

For example, at standard temperature and pressure (0°C and 1 atmosphere), one mole of any ideal gas occupies 22.4 liters, known as the standard molar volume. However, the molar volume can vary greatly with different conditions, which is why we must use the Ideal Gas Law equation to calculate it for different scenarios, such as solving for the volume in the given exercise.

The Ideal Gas Law formula, \(V = \frac{nRT}{P}\), shows that the volume (V) is directly proportional to the amount of substance (n) in moles and the absolute temperature (T), and inversely proportional to the pressure (P). Understanding how these variables influence the molar volume is key to mastering problems involving gases.

Temperature is a fundamental parameter in the study of gases, and the absolute temperature scale, measured in Kelvins (K), lays the groundwork for gas laws. It starts at absolute zero, the theoretical lowest possible temperature, where all molecular motion ceases.

These temperatures cannot be negative, because Kelvins are an absolute measure of thermal energy. To convert Celsius to Kelvin, the equation, \(K = ^\circ{}C + 273.15\), is used. For instance, in solving for the absolute temperature in the exercise provided, we use Kelvin to ensure that the temperature values are scientifically reasonable.

Since the Ideal Gas Law equations depend on absolute temperature, it is crucial to convert any Celsius temperatures into Kelvins to achieve accurate results. This reflects the direct relationship between temperature and other properties of gases: pressure, volume, and the number of moles.

Gas pressure is the force exerted by gas molecules as they collide with the walls of their container. It is dependent on the number of gas molecules, their speed, and the size of the container. Gas pressure can be measured in various units such as atmospheres (atm), Pascals (Pa), and torr, among others.

Pressure is a crucial variable in the Ideal Gas Law equation and can fundamentally alter gas behavior. When measuring pressure, it is important to convert it to the appropriate units before performing calculations, as shown in the exercise where torr is converted to atmospheres and kilopascals to atmospheres.

Pressure is inversely proportional to volume, assuming the number of moles and temperature are constant. As demonstrated in the exercise, if you increase the pressure exerted on a fixed amount of gas, the volume decreases, a concept central to understanding gas laws.

In chemistry, the mole is a unit of measurement for the amount of substance. It is one of the seven base SI units and is defined as containing exactly \(6.022 \times 10^{23}\) entities (Avogadro's number). When the context is the Ideal Gas Law, 'moles of gas' refers to the number of actual gas particles, whether they are atoms or molecules, contained in a sample.

The number of moles of gas is symbolized by the variable 'n' in the Ideal Gas Law and is directly proportional to the volume and temperature, while being inversely proportional to the pressure. Understanding the quantity of gas and how to calculate it, as illustrated in the final part of the exercise, is crucial in stoichiometry and for predicting how gases will respond to changes in physical conditions.