Problem 80 - 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 number of molecules of NO2 in the room under the given conditions is approximately \( 1.91 \times 10^{19} \) molecules.

Step 1: Convert the concentration of NO2 from ppm to a fraction

To convert the concentration of NO2 from ppm to a fraction, divide the concentration (0.016 ppm) by 1,000,000 (since ppm stands for parts per million): \[ \frac{0.016}{1,000,000} = 1.6 \times 10^{-8} \]

Step 2: Calculate the partial pressure of NO2

To calculate the partial pressure of NO2, multiply the atmospheric pressure (755 torr) by the fraction obtained in Step 1: \[ 755 \text{ torr} \times 1.6 \times 10^{-8} = 1.21 \times 10^{-5}\text{ torr} \]

Step 3: Find the number of moles of air in the room

We can use the ideal gas law (PV = nRT) to find the number of moles of air in the room. First, we need to convert the room dimensions from feet to meters: \[ \frac{15 \text{ ft} \times 14 \text{ ft} \times 8 \text{ ft}}{3.2808^3\frac{\text{m}^3}{\text{ft}^3}} = 47.89 \text{ m}^3 \] Next, we'll convert the pressure from torr to Pa: \[ \frac{755 \text{ torr}}{0.007501 \frac{\text{torr}}{\text{Pa}}} = 100662.86 \text{ Pa} \] Now, we'll plug our values into the ideal gas law, rearranged for the number of moles: \[ n = \frac{PV}{RT} = \frac{ (100662.86 \text{ Pa})(47.89 \text{ m}^3)}{(8.314 \text{ J/(mol·K)})(20 + 273.15 \text{ K})} = 1984.99 \text{ moles} \]

Step 4: Calculate the number of moles of NO2 in the room

To find the number of moles of NO2, multiply the fraction obtained in Step 1 by the total number of moles of air in the room: \[ 1.6 \times 10^{-8} \times 1984.99 \text{ moles} = 3.18 \times 10^{-5} \text{ moles of } NO_2 \]

Step 5: Convert the moles of NO2 to molecules of NO2

To convert the moles of NO2 to molecules, multiply the number of moles by Avogadro's number (6.022 x 10^23 molecules/mol): \[ 3.18 \times 10^{-5} \text{ moles of } NO_2 \times 6.022 \times 10^{23} \frac{\text{molecules}}{\text{mol}} = 1.91 \times 10^{19} \text{ molecules of } NO_2 \] The number of molecules of NO2 in the room under the given conditions is approximately \( 1.91 \times 10^{19} \) molecules.

Key Concepts

Partial Pressure

When we talk about partial pressure, we are discussing the contribution of a single type of gas to the total pressure in a mixture of gases. Essentially, each gas in a mixture exerts its own pressure as if it were alone in the volume, and this is its partial pressure.

To find the partial pressure of \(O_2\) in our example, it's important to first convert the concentration provided in parts per million (ppm) to a fraction by dividing by 1,000,000. This fraction represents the proportion of \(O_2\) compared to the total air. Then, we multiply the total atmospheric pressure by this fraction to get the partial pressure.

For instance, with an atmospheric pressure of 755 torr and a \(O_2\) concentration of 0.016 ppm, the partial pressure is calculated as: \[ 755 \, ext{torr} \times 1.6 \times 10^{-8} = 1.21 \times 10^{-5} \, ext{torr} \] Understanding this fraction and multiplication helps to comprehend how each gas component contributes to the total pressure in a mixture.

Ideal Gas Law

The Ideal Gas Law is one of the essential equations in chemistry and physics, helping us to understand the behavior of gases under various conditions. The equation is written as \( PV = nRT \), where:

• \(P\) is the pressure of the gas
• \(V\) is the volume of the gas
• \(n\) is the number of moles of gas
• \(R\) is the universal gas constant (8.314 \text{ J/(mol·K)})
• \(T\) is the temperature in Kelvin

To find the moles of air in the room, we rearrange this formula to solve for \(n\) as follows: \[ n = \frac{PV}{RT} \]

In the exercise, we need to convert all measurements into compatible units. For pressure, convert torr to pascals; for volume, change cubic feet to cubic meters; and for temperature, use Kelvin by adding 273.15 to the Celsius value. Plugging these values into the formula allows us to calculate the total moles of air, crucial for further calculations in the problem.

Moles and Molecules

The terms moles and molecules might sound confusing, but they are fundamental in chemical studies. A mole is a unit representing a specific number of particles, usually atoms or molecules, equivalent to Avogadro's number, which is approximately \(6.022 \times 10^{23}\).

In our context, converting moles to molecules involves multiplying the number of moles by Avogadro's number. More intuitively, one can think of moles as a bridge between measurable quantities of substances and the number of molecules or atoms.

For \(\mathrm{NO}_2\), once we determine the number of moles in the room, we can easily compute the total number of molecules by using: \[ 3.18 \times 10^{-5} \, \text{moles} \times 6.022 \times 10^{23} \, \frac{\text{molecules}}{\text{mol}} = 1.91 \times 10^{19} \, \text{molecules} \] This calculation helps in understanding how many actual \(\mathrm{NO}_2\) molecules exist in the given space, making the concept of moles tangible and applicable.