Problem 78 - Chemistry: The Central Science in SI Units, Global Edition

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 \(101 \mathrm{kPa} .(\mathbf{b})\) 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

The partial pressure of \(\mathrm{NO}_2\) is approximately \(\mathrm{1.01 \times 10^{-3}\ \mathrm{kPa}\). There are approximately \(\mathrm{4.88 \times 10^{22}}\) molecules of \(\mathrm{NO}_2\) present in the room under the given conditions.

Step 1: Part (a): Calculate the partial pressure of \(\mathrm{NO}_{2}\)

Since we are given the concentration of \(\mathrm{NO}_2\) in the air (0.010 ppm) and the atmospheric pressure (101 kPa), we can use the following formula to find the partial pressure of \(\mathrm{NO}_2\): Partial pressure of \(\mathrm{NO_2}\) = (Concentration of \(\mathrm{NO_2}\)) × (Total atmospheric pressure) First, we need to convert the concentration of \(\mathrm{NO}_2\) from ppm (parts per million) to a ratio: \(0.010 \ ppm = 0.010 \times 10^{-6}\) Now, we can find the partial pressure of \(\mathrm{NO}_2\): Partial pressure of \(\mathrm{NO_2}\) = \( 0.010 \times 10^{-6} \times 101\ \mathrm{kPa}\) Partial pressure of \(\mathrm{NO_2}\) = \(\mathrm{1.01 \times 10^{-3} \ kPa}\)

Step 2: Part (b): Calculate the number of molecules of \(\mathrm{NO}_{2}\)

To calculate the number of molecules of \(\mathrm{NO}_2\) in the room, we need to find the number of moles of \(\mathrm{NO}_2\) present and then convert that to the number of molecules using Avogadro's number. We can start by finding the volume of the room: Volume = length × width × height = \(10\ \mathrm{m} \times 8\ \mathrm{m} \times 2.50\ \mathrm{m} = 200\ \mathrm{m^3}\) Next, we need to convert the volume to liters: \(200\ \mathrm{m^3} \times \frac{1000\ \mathrm{L}}{1\ \mathrm{m^3}} = 200,000\ \mathrm{L}\) Now, we can use the ideal gas law formula to find the number of moles of \(\mathrm{NO_2}\) present in the room: \(PV = nRT\) We will first isolate n (moles) in the equation: \(n = \frac{PV}{RT}\) Where: P = Partial pressure of \(\mathrm{NO_2}\) = \(\mathrm{1.01 \times 10^{-3} \ kPa}\) V = Volume of the room = \(200,000\ \mathrm{L}\) R = Ideal gas constant = \(\mathrm{8.314\ \frac{J}{mol \cdot K}}\) (we will need to convert kPa to J/L) T = Temperature = \(25^{\circ}\mathrm{C} = 298.15\ \mathrm{K}\) First, let's convert the pressure in kPa to J/L: \(\mathrm{1.01 \times 10^{-3} \frac{kPa}{L}} \times \frac{1000\ \mathrm{J}}{1\ \mathrm{kPa \cdot L}} = 1.01 \times 10^{-3} \ \frac{J}{L}\) Now we can substitute the known values into the formula: \(n = \frac{(1.01 \times 10^{-3}\ \mathrm{J/L})(200,000\ \mathrm{L})}{(8.314\ \mathrm{J/mol \cdot K})(298.15\ \mathrm{K})}\) Solving for n: \(n \approx 0.0812\ \mathrm{moles}\) Now, we can use Avogadro's number to convert moles to molecules: Number of molecules = (moles) × (Avogadro's number) = \(0.0812\ \mathrm{moles} \times 6.022 \times 10^{23}\ \mathrm{molecules/mol}\) Number of molecules of \(\mathrm{NO_2}\) = \(\mathrm{\approx 4.88 \times 10^{22} \ molecules}\)

Key Concepts

Partial Pressure

Partial pressure is the pressure that a single gas in a mixture of gases would exert if it occupied the entire volume alone. To calculate the partial pressure of a given gas, we need to know the total pressure of the gas mixture and the concentration of the desired gas. In this exercise, the concentration of nitrogen dioxide (\(\text{NO}_2\)) is given in parts per million (ppm).
To convert ppm to a ratio, we utilize the fact that 0.010 ppm is the same as 0.010 parts of \(\text{NO}_2\) per 1,000,000 parts of air. So, 0.010 ppm can be rewritten as 0.010 × 10⁻⁶. By multiplying this ratio with the total atmospheric pressure (101 kPa), we derive the partial pressure of \(\text{NO}_2\). This simple multiplication reflects how pressure is shared among gases in a mixture.

Ideal Gas Law

The Ideal Gas Law is vital for calculating the relationships between pressure (P), volume (V), number of moles (n), and temperature (T) of a gas. The law is mathematically expressed as:\(PV = nRT\).
In this scenario, we rearrange the formula to solve for the number of moles (\(n\)) of \(\text{NO}_2\) by isolating it on one side:\(n = \frac{PV}{RT}\).

• \(P\) is the partial pressure in Joules per Liter (J/L).
• \(V\) is the room volume converted to liters \((200,000\ \text{L})\).
• \(R\) represents the ideal gas constant \((8.314\ \text{J/mol}\cdot\text{K})\).
• \(T\) is temperature in Kelvin (\(298.15\ \text{K})\).

By plugging these values into the equation, we determine the number of moles of \(\text{NO}_2\) present in the room.

Avogadro's Number

Avogadro's Number is a crucial concept in chemistry, representing the number of constituent particles, usually atoms or molecules, in one mole of a substance. The value is approximately:\(6.022 \times 10^{23}\) particles/mole.
In this exercise, after determining that there are approximately 0.0812 moles of \(\text{NO}_2\) in the room, Avogadro's Number is employed to convert this figure into the actual number of molecules. By multiplying the number of moles by Avogadro's constant, we calculate the actual molecule count, ensuring we comprehend the vast scale of the microscopic world in practical terms.

Molecular Concentration

Molecular concentration refers to the amount of a given molecule present within a specific volume or environment. It's commonly expressed in units such as ppm, molarity, or partial pressure.
For the task at hand, we start with a molecular concentration of \(\text{NO}_2\) measured in ppm. This unit indicates the proportion of \(\text{NO}_2\) relative to one million parts of air. By converting this concentration into a usable ratio (0.010 × 10⁻⁶ for our calculations), we can compute the \(\text{NO}_2\) partial pressure in kPa and determine its impact within the specified room volume.
Understanding molecular concentration assists chemists in predicting how gases behave and react in varied conditions and applications.