Question

(a) Find the number of molecules in \(1.00 \mathrm{~m}^{3}\) of air at \(20.0^{\circ} \mathrm{C}\) and at a pressure of \(1.00 \mathrm{~atm} .(b)\) What is the mass of this volume of air? Assume that \(75 \%\) of the molecules are nitrogen \(\left(\mathrm{N}_{2}\right)\) and \(25 \%\) are oxygen \(\left(\mathrm{O}_{2}\right)\).

Short answer

The number of molecules in \(1.00 \mathrm{~m}^{3}\) of air at \(20.0^{\circ} \mathrm{C}\) and at a pressure of \(1.00 \mathrm{~atm}\) will depend on the values gotten from step 2. Also, the mass of this volume of air will be the total gotten from step 3

Step-by-step solution

The Ideal Gas Law

Use the ideal gas law which is given as: \(PV = nRT\) 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. Rewrite the formula to show n (the number of moles) = \(PV/RT\). Now, substitute the given values into the formula. Remember that the temperature must be in Kelvin, so \(20.0^{\circ} \mathrm{C}\) is equivalent to 293.15 K.

Convert number of moles to number of molecules

Now, using the Avogadro's number (\(6.022 \times 10^{23}\) molecules/mole), convert the number of moles calculated from step 1 to number of molecules by multiplying the number of moles by Avogadro's number.

Calculation of the mass

To find the mass, first find the total moles of Nitrogen and Oxygen gases using their percentage compositions, then, multiply the number of moles by their respective molar masses. Add the two values to get the total mass.

Key concepts

Avogadro's Number

Avogadro's number, also known as Avogadro's constant, is a fundamental figure in chemistry that provides a link between the micro world of atoms and molecules and the macro world of grams and liters. It is defined as the number of particles (usually atoms or molecules) in one mole of substance, and is equal to approximately 6.022 x 10^23 particles per mole. This constant is named after the Italian scientist Amedeo Avogadro, who first proposed the idea.

Understanding Avogadro's number is crucial for converting between the number of moles of a substance and the number of its constituent particles. Whether dealing with a gas, solid, or liquid, the concept of moles provides a bridge between the mass of a substance (measured in grams) and its amount in terms of particles. For instance, saying you have '1 mole of nitrogen gas' is synonymous with saying you have '6.022 x 10^23 molecules of nitrogen gas', thanks to Avogadro's number.

Molar Mass

The molar mass of a substance is the mass in grams of one mole of that substance. It is a measure that links the microscale (molecules and atoms) to the macroscale (grams). The molar mass is typically expressed in units of grams per mole (g/mol) and can be found by adding up the atomic masses of the elements in a compound as given by the periodic table.

For example, the molar mass of water (H2O) is approximately 18 g/mol, because each hydrogen atom has an approximate atomic mass of 1 g/mol and oxygen has an atomic mass of approximately 16 g/mol, so adding two hydrogen atoms (2 g/mol) to one oxygen atom (16 g/mol) equals 18 g/mol. To calculate the mass of a specific number of moles of a substance, simply multiply the number of moles by the molar mass of the substance. This calculation is vital in chemical equations and reactions, making molar mass an essential concept in chemistry and related sciences.

Gas Volume and Pressure

The relationship between the volume and pressure of a gas is described by the Ideal Gas Law, which combines several simple gas laws into one. At a fixed temperature and amount of gas (number of moles), the product of the pressure (P) and volume (V) of a gas is a constant. This relation is mathematically expressed in the Ideal Gas Law as: \( PV = nRT \). Here, \(n\) denotes the number of moles of the gas, \(R\) is the ideal (or universal) gas constant, and \(T\) is the temperature in kelvin.

The Ideal Gas Law indicates that for a given amount of gas at constant temperature, if the volume of the gas increases, the pressure decreases, and vice versa – this behavior is known as Boyle's Law. Meanwhile, keeping the pressure constant, as the temperature of a gas increases, its volume increases accordingly, known as Charles's Law. Both these fundamental behaviors of gases are encompassed in the Ideal Gas Law, which is used extensively in chemistry and physics to predict the behavior of gases under different conditions.