Problem 69 - Chemistry: The Central Science

Question

The estimated average concentration of \(\mathrm{NO}_{2}\) in air in the United States in 2006 was \(0.016\) ppm. (a) Calculate the partial pressure of the \(\mathrm{NO}_{2}\) in a sample of this air when the atmospheric pressure is 755 torr \((99.1 \mathrm{kPa}) .\) (b) How many molecules of \(\mathrm{NO}_{2}\) are present under these conditions at \(20^{\circ} \mathrm{C}\) in a room that measures \(15 \times 14 \times 8 \mathrm{ft}\) ?

Step-by-Step Solution

Answer

The partial pressure of NO₂ in the sample of air is approximately \(1.208 \times 10^{-5}\) torr. In a room measuring \(15 \times 14 \times 8\) ft at \(20^{\circ}C\), there are approximately \(1.94 \times 10^{21}\) molecules of NO₂ present under these conditions.

Step 1: Calculate the partial pressure of NO2

To calculate the partial pressure of a gas, we can use the formula: Partial Pressure of Gas = Mole Fraction × Total Pressure Mole fraction is given by concentration in ppm (parts per million), so we need to divide the concentration by 1,000,000. Mole Fraction of NO2 = 0.016 ppm / 1,000,000 = 1.6 × 10^{-8} Now, we can find the partial pressure using the given atmospheric pressure: Partial Pressure of NO2 = 1.6 × 10^{-8} (mole fraction) × 755 torr (atmospheric pressure) = 1.208 × 10^{-5} torr

Step 2: Convert the room dimensions to meters

The room dimensions are given in feet. To make calculations easier, we'll convert them to meters using the conversion factor: 1 ft = 0.3048 meters. Dimensions in meters: Length = 15 ft × 0.3048 = 4.572 m Width = 14 ft × 0.3048 = 4.2672 m Height = 8 ft × 0.3048 = 2.4384 m Now we can find the volume of the room in cubic meters: Room volume = Length × Width × Height = 4.572 m × 4.2672 m × 2.4384 m = 47.285 m^3

Step 3: Apply the Ideal Gas Law

Now we can use the Ideal Gas Law to find the number of molecules of NO2 in the room: n = (P × V) / (R × T) where n = number of moles of NO2 P = partial pressure of NO2 (Step 1 result) V = volume of the room (Step 2 result) R = ideal gas constant (8.314 J/mol·K) T = temperature in Kelvin (20°C + 273.15 = 293.15 K) First, convert the partial pressure from torr to Pascals: 1.208 × 10^{-5} torr × 133.322 = 1.6106 Pa Now, use the Ideal Gas Law equation: n = (1.6106 Pa × 47.285 m^3) / (8.314 J/mol·K × 293.15 K) = 0.00322 moles of NO2

Step 4: Calculate the number of molecules of NO2

Finally, to find the number of molecules in the room, use Avogadro's number (6.022 × 10^23 particles/mol): NO2 molecules = 0.00322 moles × 6.022 × 10^23 particles/mol = 1.938 × 10^21 molecules of NO2 So, there are approximately 1.94 × 10^21 molecules of NO2 in the room given the conditions and dimensions.

Key Concepts

Partial Pressure

When we talk about gases, we often deal with mixtures containing multiple gases. Each gas in a mixture exerts its own pressure, known as partial pressure. The partial pressure of a gas can be calculated using the formula: \(\text{Partial Pressure} = \text{Mole Fraction} \times \text{Total Pressure}\). This means that the contribution of a specific gas to the total pressure depends on how much of that gas is present compared to the whole mixture.
For instance, in the exercise, the concentration of \(\mathrm{NO}_2\) is given as 0.016 ppm. To find its mole fraction, we divide this by 1,000,000, resulting in 1.6 × 10^{-8}. This tiny mole fraction reflects the very small amount of \(\mathrm{NO}_2\) present.
We then multiply this mole fraction by the total atmospheric pressure (755 torr) to find the partial pressure of \(\mathrm{NO}_2\). This calculation provides us with the pressure \(\mathrm{NO}_2\) contributes on its own, helping us understand its behavior in the mixture.

Mole Fraction

The concept of mole fraction helps us understand the composition of a gas mixture. It tells us what fraction of the total amount of gas is made up of a specific gas. In simpler terms, it's like finding out what proportion of the entire set a single component represents.
In our exercise, we deal with \(\mathrm{NO}_2\)'s concentration, which is measured in parts per million (ppm). To convert this into a mole fraction, you divide by 1,000,000, because ppm stands for ‘per million’. Thus, a concentration of 0.016 ppm gives a mole fraction of 1.6 × 10^{-8}.
Computing the mole fraction is essential as it serves as a stepping-stone to determine the partial pressure of a single gas in a mixture. It's an important concept that makes it easier to work with gas behaviors using equations such as the Ideal Gas Law.

Avogadro's Number

Avogadro's Number is a vital constant in chemistry that provides a bridge between the atomic scale and the macroscopic scale. It allows us to translate between moles and number of molecules. This number, approximately \(6.022 \times 10^{23}\) particles per mole, denotes the quantity of atoms, ions, or molecules in one mole of substance.
In the exercise, we calculated the moles of \(\mathrm{NO}_2\) using the Ideal Gas Law and the partial pressure from earlier calculations. Once we have the amount in moles (here, 0.00322 moles), we multiply by Avogadro’s Number to find how many individual molecules are present. This yields approximately \(1.938 \times 10^{21}\) molecules of \(\mathrm{NO}_2\), showcasing the vast number of molecules even a minuscule amount like a mole can consist of.
This transformation is crucial for understanding the scale at which chemical processes occur, and it helps chemists and students alike make sense of reactions in a comprehensible manner.

Gas Constant

The gas constant \(R\) is a key component in the Ideal Gas Law. This universal constant connects the physical properties of gases, namely pressure, volume, and temperature, to the amount of gas via the equation: \(n = \frac{P \times V}{R \times T}\), where \(n\) is the number of moles.
In calculations, \(R\) is often used in different units depending on the pressure and volume units involved. For this exercise, we use \(R = 8.314\) J/mol·K to accommodate pressure in Pascals and volume in cubic meters.
Using the gas constant in the Ideal Gas Law allows us to precisely determine the number of moles for \(\mathrm{NO}_2\) in the room, which is initially in terms of pressure and volume. The constant \(R\) is critical because it ensures that the relationship between pressure, volume, temperature, and moles is accurate and consistent across various conditions and systems.