Problem 68 - University Physics with Modern Physics

Question

What is the total mass of all the oxygen molecules in a cubic meter of air at normal temperature \(\left(25^{\circ} \mathrm{C}\right)\) and pressure \(\left(1.01 \cdot 10^{5} \mathrm{~Pa}\right) ?\) Note that air is about \(21 \%\) (by volume) oxygen (molecular \(\mathrm{O}_{2}\) ), with the remainder being primarily nitrogen (molecular \(\mathrm{N}_{2}\) ).

Step-by-Step Solution

Answer

Answer: The total mass of oxygen molecules in a cubic meter of air at normal temperature and pressure is approximately 273.6 grams.

Step 1: Calculate the number of moles of air in a cubic meter

To calculate the number of moles in a cubic meter of air, we can use the ideal gas equation: \(PV = nRT\) where P is the pressure (in Pa), V is the volume (in m³), n is the number of moles, R is the ideal gas constant (8.314 J/mol·K), and T is the temperature (in Kelvin) First, we need to convert the temperature from Celsius to Kelvin: \(T = 25^{\circ}C + 273.15 = 298.15 K\) Now, we can plug in the known values and solve for n: \((1.01 \cdot 10^5 Pa) \cdot (1 m^3) = n \cdot (8.314 J/mol·K) \cdot (298.15 K)\) Solving for n, we get: \(n = \dfrac{1.01 \cdot 10^5 Pa \cdot 1 m^3}{8.314 J/mol·K \cdot 298.15 K} \approx 40.73 \, mol\)

Step 2: Find the proportion of oxygen in the air

We are given that air is 21% oxygen by volume. Therefore, to find the number of moles of oxygen in 1 cubic meter of air, we can multiply the total moles (40.73) by 0.21 (21%): \(n_{O_2} = 40.73 \, mol \cdot 0.21 = 8.55 \, mol\)

Step 3: Calculate the mass of oxygen in a cubic meter of air

Now that we have the number of moles of oxygen, we can calculate the mass using the molecular weight of oxygen (O₂): Molecular weight of O₂ = 2 * molecular weight of O = 2 * 16 g/mol = 32 g/mol. The mass of oxygen (m_O₂) can be calculated using the number of moles and the molecular weight: \(m_{O_2} = n_{O_2} \cdot MW_{O_2} = 8.55 \, mol \cdot 32 \dfrac{g}{mol} = 273.6 g\) So, the total mass of oxygen molecules in a cubic meter of air at normal temperature and pressure is approximately 273.6 grams.