Problem 78 - Chemistry The Central Science In Si Units

Question

The estimated average concentration of \(\mathrm{NO}_{2}\) in air in the United States in 2015 was 0.010 ppm. (a) Calculate the partial pressure of the \(\mathrm{NO}_{2}\) in a sample of this air when the atmospheric pressure is \(101 \mathrm{kPa}\). (b) How many molecules of \(\mathrm{NO}_{2}\) are present under these conditions at \(25^{\circ} \mathrm{C}\) in a room that measures \(10 \mathrm{~m} \times 8 \mathrm{~m} \times 2.50 \mathrm{~m} ?\)

Step-by-Step Solution

Answer

(a) Partial pressure of \(\mathrm{NO}_2\) is \(1.01 \times 10^{-3} \mathrm{Pa}\). (b) There are approximately \(4.92 \times 10^{19}\) molecules of \(\mathrm{NO}_2\) in the room.

Step 1: Understand the Problem

We need to find two things: (a) the partial pressure of \(\mathrm{NO}_2\) in air and (b) the number of \(\mathrm{NO}_2\) molecules in a given volume of air at a specific temperature and total pressure.

Step 2: Partial Pressure Calculation

The partial pressure of a gas in a mixture can be calculated using the formula: \[ P(\text{gas}) = \text{mole fraction} \times \text{total pressure} \]The concentration of \(\mathrm{NO}_2\) in ppm is equivalent to its mole fraction in the atmosphere divided by 1,000,000. Thus, if the concentration is 0.010 ppm, the mole fraction of \(\mathrm{NO}_2\) is: \[ \frac{0.010}{1,000,000} = 1.0 \times 10^{-8} \]The partial pressure of \(\mathrm{NO}_2\) is then: \[ P(\mathrm{NO}_2) = 1.0 \times 10^{-8} \times 101,000 \, \mathrm{Pa} = 1.01 \times 10^{-3} \mathrm{Pa} \]

Step 3: Determine the Volume of the Room

Calculate the volume of the room using its dimensions: \[ V = \text{length} \times \text{width} \times \text{height} = 10 \times 8 \times 2.5 = 200 \mathrm{~m}^3 \]

Step 4: Convert Temperature to Kelvin

Convert the given temperature to Kelvin: \[ T(K) = 25^\circ C + 273.15 = 298.15 \mathrm{~K} \]

Step 5: Calculate Number of Molecules using Ideal Gas Law

The number of moles of \(\mathrm{NO}_2\) can be found using the ideal gas law: \[ PV = nRT \]We rearrange for moles \( n \): \[ n = \frac{PV}{RT} \]Plug in the values, using \( R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1} \): \[ n = \frac{1.01 \times 10^{-3} \times 200}{8.314 \times 298.15} \approx 8.17 \times 10^{-5} \text{ moles} \]Convert moles to molecules by multiplying by Avogadro's number \( 6.022 \times 10^{23} \): \[ \text{Number of molecules} = 8.17 \times 10^{-5} \times 6.022 \times 10^{23} \approx 4.92 \times 10^{19} \text{ molecules} \]

Step 6: Compile the Results

(a) The partial pressure of \(\mathrm{NO}_2\) is \(1.01 \times 10^{-3} \mathrm{Pa}\). (b) There are approximately \(4.92 \times 10^{19}\) molecules of \(\mathrm{NO}_2\) in the room.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and the amount of gas in moles. It is given by the formula \( PV = nRT \) where:

• \( P \) represents pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant \( (8.314 \, \text{J mol}^{-1} \text{ K}^{-1}) \)
• \( T \) is the temperature in Kelvin

By rearranging this equation, students can solve for any unknown variable if the others are given.
For partial pressure calculations, the law helps determine the amount of gas molecules in a given space and is crucial in scenarios where gases behave ideally.

Mole Fraction

In a mixture of gases, the mole fraction represents the ratio of the number of moles of a particular gas to the total number of moles of all gases present.The formula to calculate the mole fraction \( X \) of a gas is:

• \( X = \frac{n_{gas}}{n_{total}} \)

For computational purposes, it is often used to find the fraction of a gas in terms of its partial pressure using the relation:

• \( P_{gas} = X \times P_{total} \)

Here, the mole fraction is a useful bridge to convert ppm (parts per million) to a pressure unit like Pascal in calculations, as seen in the exercise where 1 ppm is equivalent to a mole fraction of \( 1.0 \times 10^{-6} \). This understanding allows for a simple conversion from concentration in air to its effects on measurable pressure.

Avogadro's Number

Avogadro's number is a key constant in chemistry that defines the number of atoms, ions, or molecules in a mole of a substance, specifically \(6.022 \times 10^{23}\).This allows chemists to relate quantities of substances on a microscopic scale to those on a macroscopic scale.
For gases, multiplying the number of moles by Avogadro's number gives the total number of gas molecules.
It's particularly useful when determining how many molecules are present in a certain room, as required in our example exercise.
This bridge between moles and actual molecular count helps make sense of everything from simple gas volumes to complex reactions in chemistry.

Room Volume Calculation

Calculating the volume of a room is a straightforward process requiring the multiplication of its length, width, and height. For a room with dimensions given in meters, the volume \( V \) is calculated as:

• \( V = \text{length} \times \text{width} \times \text{height} \)

This volume helps in calculating the total amount of gas present at a specific temperature and pressure, which is vital in applications like controlled atmospheres and pollution measurement.
In our exercise, measuring the room volume provides a key piece of the puzzle for determining the number of \(\mathrm{NO}_2\) molecules. This calculation helps lay the foundation for various chemical and physical experiments in laboratories and real-world applications.