(a) A \(125-\mathrm{mL}\) flask contains argon at \(1.30\) atm and \(77^{\circ} \mathrm{C}\). What amount of Ar is present (in moles)? (b) A \(120 .-\mathrm{mL}\) flask contains \(2.7 \mu \mathrm{g}\) of \(\mathrm{O}_{2}\) at \(17^{\circ} \mathrm{C}\). What is the pressure (in Torr)? (c) A 20.0-L flask at \(215 \mathrm{~K}\) and 20. Torr contains nitrogen. What mass of nitrogen is present (in grams)? (d) A 16.7-g sample of krypton exerts a pressure of \(1.00 \times 10^{2} \mathrm{~m}\) Torr at \(44^{\circ} \mathrm{C}\). What is the volume of the container (in liters)? (e) A \(2.6-\mu \mathrm{L}\) ampoule of xenon has a pressure of \(2.00\) Torr at \(15^{\circ} \mathrm{C}\). How many Xe atoms are present?

The mole is a fundamental unit in chemistry that provides a bridge between the atomic and macroscopic worlds. One mole of any substance contains exactly as many entities (such as atoms or molecules) as there are atoms in exactly 12 grams of carbon-12. This connects directly to the Ideal Gas Law which states that the volume of a gas is directly proportional to the number of moles of gas present when the pressure and temperature are held constant. Calculating the moles of a gas involves using the Ideal Gas Law, represented by the equation
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvins.For instance, to find the number of moles of argon in a flask, one would rearrange the Ideal Gas Law as
n = PV / RT
after converting all measurements to the appropriate units (pressure in atm, volume in liters, temperature in Kelvins). Understanding the mole concept is crucial in chemistry as it enables us to count particles by weighing.

The Standard Gas Equation is synonymous with the Ideal Gas Law, and it is a cornerstone of gas calculations in chemistry. It establishes a relationship between the four state variables of a gas: pressure (P), volume (V), moles of gas (n), and temperature (T). The equation takes the form
PV = nRT
where R is the universal gas constant with a value of 0.0821 L·atm/K·mol. For example, if we need to find the pressure exerted by oxygen in a flask, we can rearrange this equation to calculate P while considering the conditions of temperature and volume under standard pressure and temperature conditions. The use of this formula requires that all units are consistent, and any necessary conversions are made before calculation. The Standard Gas Equation allows for the prediction of the behavior of gases under various conditions and is fundamental to both experimental and theoretical chemistry.

Avogadro's Number is a key constant in chemistry, representing the number of particles in one mole of a substance. It is defined as
6.022 \( \times \) 10^23 particles/mol,
and it allows chemists to convert between the number of atoms or molecules and the amount of substance in moles. This conversion is crucial when dealing with gas laws, as it enables one to determine the exact number of gas particles in a given volume, under defined conditions of temperature and pressure. For example, when calculating the number of xenon atoms in a sample, after finding the moles of xenon using the Ideal Gas Law, we multiply
n \( \times \) Avogadro's Number
to obtain the total number of individual atoms. Avogadro's Number ultimately bridges the macroscopic scale of laboratory measures with the microscopic scale of atoms and molecules.

Molar mass is the mass of one mole of a particular substance and is expressed in grams per mole (g/mol). It is an intrinsic property of each element or compound, being equivalent to the atomic or molecular mass reported in unified atomic mass units (u) on the periodic table. The importance of molar mass in the context of gas equations lies in its ability to convert between mass and moles of a substance. For instance, to calculate the mass of nitrogen in a flask, we find the number of moles using the Ideal Gas Law, and then, considering the molar mass of nitrogen as 28.02 g/mol, we solve for the mass with
mass = n \( \times \) molar mass.
Molar mass is essential in stoichiometry for predicting the outcomes of chemical reactions and is central to all quantitative analyses in chemistry.