Problem 13 - Chemistry The Central Science In Si Units

Question

The Environmental Protection Agency (EPA) has established air quality standards. For ozone \(\left(\mathrm{O}_{3}\right)\), the 8 -hour average concentration permitted under the standards is 0.085 parts per million (ppm). (a) Calculate the partial pressure of ozone at 0.085 ppm if the atmospheric pressure is \(100 \mathrm{kPa}\). (b) How many ozone molecules are in \(1.0 \mathrm{~L}\) of air? Assume \(T=25^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

(a) 0.0085 Pa; (b) Approximately \(2.06 \times 10^{15}\) ozone molecules.

Step 1: Understand Partial Pressure

The partial pressure of a gas in a mixture can be calculated using its concentration and the total pressure. Here, we have the concentration in parts per million (ppm) and the atmospheric pressure is given as 100 kPa.

Step 2: Conversion of Concentration Units

Convert the ozone concentration from ppm to a fraction for easier calculation of partial pressure. Since 0.085 ppm means 0.085 parts of ozone in 1 million parts of air, it is equivalent to 0.085/1,000,000.

Step 3: Calculate Partial Pressure of Ozone

Use the equation for partial pressure: \[ P_{O_3} = rac{C}{1,000,000} \times P_{total} \]Substitute the given values: \[ P_{O_3} = \frac{0.085}{1,000,000} \times 100,000 \] Convert 100 kPa to Pa by multiplying by 1,000. Thus, \( P_{O_3} = 0.0085 \) Pa.

Step 4: Use Ideal Gas Law to Find Moles of Ozone

Use the Ideal Gas Law \( PV = nRT \) where \( P \) is the partial pressure of ozone (in Pa), \( V = 1.0 \text{ L}\), \( R = 8.314 \text{ J/(mol K)} \), and \( T = 298 \text{ K} \) (25°C = 298 K). First, convert the volume from liters to cubic meters (1 L = 0.001 m³):\[ n = \frac{P_{O_3}V}{RT} \] Substitute: \[ n = \frac{0.0085 \times 0.001}{8.314 \times 298} \] Calculate \( n = 3.42 \times 10^{-9} \text{ moles}.\)

Step 5: Calculate Number of Ozone Molecules

To find the number of molecules, multiply the moles of ozone by Avogadro's number \( 6.022 \times 10^{23} \text{ molecules/mol} \):\[ \text{Number of molecules} = 3.42 \times 10^{-9} \times 6.022 \times 10^{23} \]This gives approximately \(2.06 \times 10^{15}\) ozone molecules in 1.0 L of air.

Key Concepts

Ozone Concentration

Understanding ozone concentration in the context of air quality revolves around measuring its presence in the air we breathe. Ozone, often denoted as \( \text{O}_3 \), is a molecule composed of three oxygen atoms. It plays a dual role in the Earth's atmosphere—beneficial in the stratosphere as it blocks harmful UV rays, yet a pollutant at ground level where it can cause respiratory issues. Air quality standards, like those established by the Environmental Protection Agency (EPA), limit the levels of ozone permitted in the air to protect both environmental and human health.### Measurement in Parts Per MillionOzone concentration is often expressed in parts per million (ppm). - One part per million means one part of a substance in a million parts of the total mixture. - For ozone, 0.085 ppm indicates very low concentrations, yet it's significant enough to impact air quality and human health. This fine measurement is crucial for ensuring that air quality remains within safe limits as defined by air quality standards.

Partial Pressure Calculation

Partial pressure is a key concept in understanding how gases behave in a mixture, such as the Earth's atmosphere. The partial pressure of a gas represents the pressure it would exert if it alone occupied the entire volume.To calculate the partial pressure of ozone, given its concentration of 0.085 ppm and an atmospheric pressure of \( 100 \text{kPa} \), we follow these steps:- Convert the ozone concentration from ppm to a fraction: \( \frac{0.085}{1,000,000} \). - Multiply this fraction by the total atmospheric pressure, converted to Pascals (Pa) for consistency in unit measure (\( 100 \text{kPa} = 100,000 \text{Pa} \)).The formula becomes:\[ P_{\text{O}_3} = \frac{0.085}{1,000,000} \times 100,000 = 0.0085 \text{Pa} \]This tells us how much pressure the ozone contributes to the overall atmospheric pressure.

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry for relating the volume, pressure, and temperature of gases with their amount in moles. It makes calculating various properties of gases straightforward using the equation:\[ PV = nRT \]Where:- \( P \) is the pressure (in Pa),- \( V \) is the volume (in m³), - \( n \) is the number of moles,- \( R \) is the ideal gas constant \( (8.314 \text{ J/mol K}) \),- \( T \) is the temperature in Kelvin. When given a volume of \( 1.0 \text{ L} \) of air (which we convert to \( 0.001 \text{ m}^3 \)), we can calculate the moles of ozone using:\[ n = \frac{P_{\text{O}_3} \times V}{R \times T} \]Substituting the values: - \( P_{\text{O}_3} = 0.0085 \text{Pa} \),- \( T \) is \( 298 \text{K} \) (as \( 25^{\circ} \text{C} = 298 \text{K} \)),Calculations give \( n = 3.42 \times 10^{-9} \text{ moles} \).This result can be further used to find the number of ozone molecules by multiplying the moles by Avogadro's number, which connects macroscopic and molecular-scale observations effectively.