Question

(a) Calculate the volume occupied by \(1.00 \mathrm{~mol}\) of an ideal gas at standard conditions - that is, pressure of \(1.00 \mathrm{~atm}\) \(\left(=1.01 \times 10^{5} \mathrm{~Pa}\right)\) and temperature of \(0^{\circ} \mathrm{C}(=273 \mathrm{~K}) \cdot(b)\) Show that the number of molecules per cubic centimeter (the Loschmidt number) at standard conditions is \(2.68 \times 10^{19}\).

Short answer

The volume of 1.00 mole of an ideal gas at standard conditions is 22.4L or \(22.4 \times 10^3cm^3\). The Loschmidt number at standard conditions is \(2.68 \times 10^{19}/cm^3\).

Step-by-step solution

Finding the Gas Standard Volume

Firstly, rearrange the Ideal Gas Law to solve for V (volume): V = nRT/P. Since we are dealing with standard conditions, the temperature T should be in Kelvin. Convert \(0^{\circ} C\) to Kelvin by adding 273, making T equal to \(273K\). Plug in the values into the formula where n=1 mole, R=0.0821 L.atm/(K.mol) (this is the value of R given the units of pressure and volume), T=273 K and P =1.00 atm. Solve for V.

Calculating Loschmidt Number in Ideal Conditions

The Loschmidt number gives us the number of molecules per cubic centimeter at standard conditions. We can derive this using the result we got from Step 1 and Avogadro's number. Avogadro's number (\(6.022 \times 10^{23}\)) gives the number of particles (in this case, molecules) in one mole of a substance. Divide Avogadro's number by the volume calculated in Step 1. Note that the volume must be in cubic centimeters (1L = \(10^3cm^3\)) for the units to cancel out correctly.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and amount of an ideal gas. This relationship is expressed as PV = nRT, where P stands for pressure, V for volume, n for the amount of substance in moles, R for the ideal gas constant, and T for temperature in Kelvin.

One critical application of this law is to determine the volume occupied by a gas at a given temperature and pressure. By setting standard pressure (\(1.00 \text{atm}\)) and standard temperature (\(273K\text{, or }0^\text{C}\text{, as in the exercise}\text{, we ensure a common baseline for comparisons—often referred to as Standard Temperature and Pressure (STP).}\)

In an educational context, when helping students visualize these concepts, it's useful to draw parallels to syringes or balloons under different conditions to showcase how volume can change with fluctuations in temperature and pressure.

Loschmidt number

The Loschmidt number is the number of molecules of an ideal gas present in a specific volume under standard conditions. It's particularly useful in understanding the scale and density of gases on the molecular level. To calculate it, we first need to know the volume occupied by the gas, which we find using the Ideal Gas Law, and then use Avogadro's number to determine the number of gas molecules present.

The process involves dividing Avogadro's number by the standard molar volume of a gas to obtain a figure representative of the molecular density in air, or any gas, at STP. While the concept might seem abstract, comparing it to familiar concepts, such as the density of a crowd in a room, can aid in comprehension. Teaching about Loschmidt number helps students appreciate the molecular scale and advance their understanding of concentrations and reaction kinetics.

Avogadro's number

Avogadro's number, approximately \(6.022 \times 10^{23}\), is a fundamental constant in chemistry representing the number of entities, usually atoms or molecules, in one mole of a substance. This number bridges the gap between the macroscopic world we can measure and observe and the microscopic world of atoms and molecules.

To understand this concept, students can think of a dozen as a familiar term for counting. Just as a dozen represents 12 items, regardless of what the items are, Avogadro's number represents the number of molecules in one mole, whether those are water molecules, oxygen molecules, or any other substance. Emphasizing the scale of Avogadro's number can also be effective; imagining filling up countless balloons with tiny particles until you reach that colossal number can put it into perspective.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. This branch of chemistry plays a crucial role in ensuring that equations are balanced and that relationships between molecules during reactions are accurately represented. It helps us quantify the amount of a substance produced or required in a given reaction.

An effective way to elucidate stoichiometry is through cooking analogies, such as following a recipe. When we adjust the quantities of ingredients while maintaining the proportions to achieve the desired result, we are essentially practicing stoichiometry. It is a practical illustration of how this concept applies to real-life scenarios and is foundational for ventures ranging from cooking to pharmaceuticals and materials science.