Question

What is the total mass of all the oxygen molecules in a cubic meter of air at a temperature of \(25.0^{\circ} \mathrm{C}\) and a pressure of \(1.01 \cdot 10^{5} \mathrm{~Pa}\) ? Note that air is \(20.9 \%\) (by volume) oxygen (molecular \(\mathrm{O}_{2}\) ), with the remainder being primarily nitrogen (molecular \(\mathrm{N}_{2}\) ).

Short answer

Answer: The total mass of oxygen molecules in a cubic meter of air at the given temperature and pressure is approximately 272.96 g.

Step-by-step solution

List the given information

In this problem, we are given: - Temperature \(T = 25.0^{\circ} \mathrm{C}\) - Pressure \(P = 1.01 \cdot 10^{5} \mathrm{~Pa}\) - Volume \(V = 1 \mathrm{~m^3}\) (cubic meter of air) - Air is \(20.9 \%\) oxygen (by volume) - Molecular Oxygen \(\mathrm{O}_{2}\) molar mass: \(32 \mathrm{~g/mol}\)

Convert the temperature to Kelvin

To use the Ideal Gas Law, we need to convert the temperature from Celsius to Kelvin. We simply add 273.15 to the Celsius temperature: \(T_\mathrm{K}= T_\mathrm{C} + 273.15\) \(T_\mathrm{K} = 25.0^{\circ} \mathrm{C} + 273.15 = 298.15 \mathrm{K}\)

Determine the number of moles of air

Using the Ideal Gas Law formula (\(P \cdot V = n \cdot R \cdot T\)), we can find the total number of moles of air in \(1 \mathrm{~m^3}\). Rearrange the formula to find the number of moles, \(n\): \(n = \frac{P \cdot V}{R \cdot T}\) Here, we will use the universal gas constant, \(R = 8.314 \mathrm{~J/(mol \cdot K)}\) \(n = \frac{1.01 \cdot 10^{5} \mathrm{~Pa} \cdot 1 \mathrm{~m^3}}{8.314 \mathrm{~J/(mol \cdot K)} \cdot 298.15 \mathrm{K}} = 40.81 \mathrm{~mol}\)

Calculate the number of moles of oxygen molecules

We are given that the air is \(20.9 \%\) oxygen by volume. Since the percentage represents the mole fraction, we can calculate the number of moles of oxygen as follows: \(n_\mathrm{O_2}= 40.81 \mathrm{~mol} \cdot 20.9\% = 40.81 \mathrm{~mol} \cdot 0.209 = 8.53 \mathrm{~mol}\)

Calculate the total mass of oxygen molecules

Now we will multiply the number of moles of oxygen by the molar mass of molecular oxygen (\(\mathrm{O}_2\)) to calculate the total mass of oxygen: Mass of \(\mathrm{O}_{2} = n_\mathrm{O_2} \cdot \mathrm{Molar\, Mass\, of\, O_2}\) Mass of \(\mathrm{O}_{2} = 8.53 \mathrm{~mol} \cdot 32 \mathrm{~g/mol} = 272.96 \mathrm{~g}\) The total mass of all oxygen molecules in a cubic meter of air at the given temperature and pressure is approximately \(272.96 \mathrm{~g}\).