Problem 80 - Chemistry An Atoms First Approach

Question

Given that a sample of air is made up of nitrogen, oxygen, and argon in the mole fractions \(0.78 \mathrm{N}_{2}, 0.21 \mathrm{O}_{2},\) and \(0.010 \mathrm{Ar},\) what is the density of air at standard temperature and pressure?

Step-by-Step Solution

Answer

The density of air at standard temperature and pressure, given the mole fractions \(0.78 N_2\), \(0.21 O_2\), and \(0.010 Ar\), is approximately 1.29 g/L.

Step 1: Find the molar mass of each component

To get the molar mass of the air sample, we need to find the molar mass of each gas involved. We use the mole fraction given in the problem and use the molar masses of nitrogen, oxygen, and argon gases to have an appropriate measure. Molar mass of N2: 28.02 g/mol Molar mass of O2: 32.00 g/mol Molar mass of Ar: 39.95 g/mol

Step 2: Calculate the molar mass of air

Now, using the mole fractions and the molar masses of each gas, we find the molar mass of air. Multiply the mole fraction of each gas by its molar mass and sum them up: M_air = (0.78 * 28.02 g/mol) + (0.21 * 32.00 g/mol) + (0.010 * 39.95 g/mol) M_air ≈ 28.97 g/mol

Step 3: Recall the ideal gas law and standard conditions

To find the density of air, we need to use the ideal gas law (PV = nRT), where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature. We are given standard temperature and pressure (STP) conditions, which means: Temperature (T) = 273.15 K (0°C) Pressure (P) = 1 atm (101.325 kPa) Also, the ideal gas constant (R) = 0.0821 (L * atm / (mol * K))

Step 4: Find the number of moles per volume

Rearranging the ideal gas law formula to find n/V: n/V = P / (RT) Let's plug in the values of P, R, and T: n/V = (1 atm) / (0.0821 L * atm / (mol * K) * 273.15 K) n/V ≈ 0.0446 mol/L

Step 5: Calculate the density of air

Now let's use the calculated molar mass of air and the number of moles per volume to find density. Density = M_air * n/V Density_air ≈ (28.97 g/mol) * (0.0446 mol/L) Density_air ≈ 1.29 g/L Thus, the density of air at standard temperature and pressure is approximately 1.29 g/L.

Key Concepts

Molar Mass

The concept of molar mass is central in understanding the density of gases. Molar mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). To find the molar mass of a mixture like air, which consists of multiple gases, you need to consider the molar mass of each individual component. For air, the primary components are:

• Nitrogen ( _2"): 28.02 g/mol
• Oxygen ( _2"): 32.00 g/mol
• Argon: 39.95 g/mol

Each gas contributes to the overall molar mass of air in proportion to its mole fraction. In our example, with mole fractions given, the calculation involves multiplying the mole fraction of each gas by its respective molar mass and summing the results. This weighted sum gives us the approximate molar mass of air, which is 28.97 g/mol. This value is a key player when assessing gas density because it helps us link the mass and the volume of the gas.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, and temperature of a gas to the number of moles it contains. Expressed as \( PV = nRT \), where:

• \( P \) is pressure
• \( V \) is volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is temperature

This law is crucial for calculating densities of gases under different conditions. For instance, when working with air, you can rearrange the equation to solve for \( n/V \), the number of moles per volume: \( n/V = P/(RT) \). By substituting known values for the pressure, temperature, and the ideal gas constant under standard conditions, we obtain \( n/V \approx 0.0446 \) mol/L. This figure represents how many moles of gas are present per liter, further helping us determine the density when multiplied by the molar mass.

Standard Temperature and Pressure (STP)

Standard temperature and pressure (STP) are conditions often used as a reference point in chemistry. Knowing these conditions is essential for working with the ideal gas law as they provide a common baseline:

• Standard Temperature: 273.15 K (0° Celsius)
• Standard Pressure: 1 atm (101.325 kPa)

At STP, gases exhibit predictable behavior that simplifies calculations of properties like density. Using these conditions, we ascertain the density of air by inputting STP values into the ideal gas law. Initially calculate \( n/V \) to find the moles per volume under STP. With this uniform backdrop, it becomes straightforward to multiply by the air's molar mass, leading us to a density of approximately 1.29 g/L. By setting these standards, STP allows for consistent and reliable comparisons across different gases and scenarios.