(a) If you filled up three balloons with equal volumes of hydrogen, argon, and carbon dioxide gas, all at the same temperature and pressure, which balloon would weigh the most? the least? Explain. (b) If you filled up three balloons with equal masses of nitrogen, oxygen, and neon, all to the same volume at the same temperature, which would have the lowest pressure?

Understanding molar mass is crucial for explaining why different gases weigh different amounts under the same conditions. The molar mass (usually denoted as \( M \)) of a substance is the mass of one mole of that substance; one mole containing \( 6.022 \times 10^{23} \) entities (atoms, molecules, ions, etc.). Units for molar mass are grams per mole (g/mol).

For gases, individual molecules travel freely and fast, and heavier molecules contribute more to the overall weight of the gas. The molar masses of common gases like hydrogen (\( H_2 \)), argon (\( Ar \)), and carbon dioxide (\( CO_2 \)) differ significantly: \( H_2 \) being the lightest at approximately 2 g/mol, \( Ar \) sits in the middle at about 40 g/mol, and \( CO_2 \) is the heaviest among them at approximately 44 g/mol. When balloons are filled with equal volumes of these gases at the same temperature and pressure, the balloon with argon or carbon dioxide will weigh more than the balloon with hydrogen due to the higher molar masses of argon and carbon dioxide.

In a practical scenario, knowing the molar mass helps in calculating the mass of a gas required for chemical reactions or the amount produced as a byproduct, a common task in laboratory and industrial settings.

Gas density is a measure of the mass of gas per unit volume, often expressed in grams per liter (g/L) or kilograms per cubic meter (kg/m^3). It is directly related to the molar mass of the gas and the conditions of temperature and pressure. The density of a gas can be calculated using the Ideal Gas Law (PV = nRT) along with the equation for molar mass (M) and the relationship between moles (n) and mass (m), thus the equation \( \rho = \frac{m}{V} = \frac{nM}{V} \).

In the exercise, this concept explains why balloons filled with different gases would exhibit varying weight. Gases with higher molar masses will have higher densities under the same temperature and pressure conditions, thus heavier. It's important to note that, at standard temperature and pressure (STP), gas densities can be easily compared. However, if conditions change, so does the density, and this must be accounted for in calculations.

The concept of gas density also explains atmospheric phenomena, such as why certain pollutants may hang low to the ground, or how hot air balloons rise. It is also the secret behind some magic tricks and children's science experiments, like 'the floating balloon' trick, which uses lighter-than-air gases.

Gas pressure is the force that a gas exerts on the walls of its container, and it's measured in units such as atmospheres (atm), pascals (Pa), or pounds per square inch (psi). Intuitively, pressure is a measure of how much the gas molecules are 'pushing' against their confines, and it is a result of the countless collisions they undertake.

Using the Ideal Gas Law again, we find that pressure (\( P \)) is directly proportional to the number of moles (\( n \)) and the temperature (\( T \)) and inversely proportional to the volume (\( V \)). In the example with balloons filled with equal masses of nitrogen, oxygen, and neon, the balloon with the gas that has the lowest molar mass will have the most moles of gas (since \( n = \frac{m}{M} \)) and thus exert the highest pressure, assuming the volume is kept constant.

Understanding gas pressure is not only important for solving academic exercises, it's also pivotal in various real-life applications including weather forecasting, cooking (like in pressure cookers), and even in the medical field with the use of pressurized oxygen tanks for patients.