Question

Dry air at STP has a density of \(1.2929 \mathrm{~g} / \mathrm{L}\). Calculate the average molecular weight of air from the density.

Short answer

The average molecular weight of air is approximately 28.97 g/mol.

Step-by-step solution

Understanding STP Conditions

At Standard Temperature and Pressure (STP), the temperature is 273.15 K and the pressure is 1 atm. These conditions are important because they provide a basis for calculations involving gases.

Using Ideal Gas Law

The Ideal Gas Law is given by the formula \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is the temperature in Kelvin. At STP, \( P = 1 \) atm, and \( T = 273.15 \) K.

Rewriting the Ideal Gas Law

Rearranging the Ideal Gas Law to find the number of moles per liter (molar concentration), we have: \( n/V = P/RT \). Substituting the values at STP, \( n/V = 1/(0.0821 imes 273.15) \approx 0.0446 \) moles/L.

Relating Density to Molar Mass

Density (\( \rho \)) is defined as mass per volume. For a gas, density can also be expressed in terms of molar mass (M) and molar concentration: \( \rho = M \cdot (n/V) \). In this case, \( \rho = 1.2929 \) g/L. Therefore, \( 1.2929 = M \cdot 0.0446 \).

Solving for Molar Mass

From \( \rho = M \cdot (n/V) \), substitute \( 1.2929 \) for \( \rho \) and solve for \( M \): \( M = 1.2929/0.0446 \approx 28.97 \) g/mol.

Key concepts

STP (Standard Temperature and Pressure)

STP, or Standard Temperature and Pressure, is a conventional reference point for gas calculations. Many times in chemistry, to make things simpler and comparisons clearer, scientists use the same set of conditions for temperature and pressure.
At STP, the temperature is defined as 273.15 Kelvin, or 0 degrees Celsius, and the pressure is 1 atmosphere (atm). These conditions are used because they represent a common environment that is easy to replicate in a laboratory setting.
This standardization allows us to predict and compare how gases behave under the same conditions, simplifying calculations. When solving problems involving gases, it’s critical to know whether STP conditions are applied, as this influences the calculations and conversion factors used.

Density of Gases

Density is a measure of how much mass is contained in a given volume. For gases, density is often expressed in units of grams per liter (g/L). Because gases can expand or compress significantly with changes in temperature or pressure, knowing the conditions is crucial when measuring or comparing densities.
The density of a gas at STP is calculated by using the formula: \[ \rho = \frac{M \cdot n}{V} \]where \( \rho \) is the density, \( M \) is the molar mass, and \( n/V \) is the molar concentration.
This allows you to relate the mass of the gas to its volume. For example, the density of dry air at STP is given as 1.2929 g/L, which means every liter of air at these conditions weighs 1.2929 grams. This relationship helps to find other properties like the average molecular weight of the gas mixture if one is known.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). It is an essential property in chemistry as it links the mass of a chemical substance to the amount of moles, which is foundational to stoichiometry.
To calculate the molar mass of a gas given its density and using the Ideal Gas Law, you rearrange the density formula: \[ M = \frac{\rho}{n/V} \]For this, you first need the molar concentration \( n/V \), which you can find using the Ideal Gas Law: \( PV = nRT \). At STP, \( n/V = \frac{P}{RT} \). Substituting in known values at STP allows the calculation of molar mass.
For example, if the density of air is known, you can calculate the average molecular weight of the gases in the air by rearranging and substituting into the formula, leading to a direct computation of molar mass. In our exercise, this results in a molar mass of approximately 28.97 g/mol for air at STP.