Question

On a smoggy day in a certain city, the ozone concentration was 0.42 ppm by volume. Calculate the partial pressure of ozone (in atm) and the number of ozone molecules per liter of air if the temperature and pressure were ${20.0}^{\circ }\mathrm{C}$ and $748\mathrm{mmHg}$, respectively.

Short answer

The partial pressure of ozone is approximately $4.13\times {10}^{-7}$ atm, and there are about $1.03\times {10}^{16}$ ozone molecules per liter.

Step-by-step solution

Convert Pressure to Atmospheres

First, we need to convert the given pressure from millimeters of mercury (mmHg) to atmospheres (atm) because standard pressure units are required for further calculations. We know that: $$1extatm=760extmmHg$$ Thus, the pressure in atmospheres is: $$P=\frac{748extmmHg}{760extmmHg/atm}\approx 0.984extatm$$

Calculate Ozone Partial Pressure

The ozone concentration is given as 0.42 ppm (parts per million by volume). The partial pressure of ozone can be calculated by multiplying the total pressure by the fraction of ozone: $$P_{\text{ozone}}=\frac{0.42}{{10}^6}\times 0.984\text{ atm}\approx 4.13\times {10}^{-7}\text{ atm}$$

Convert Temperature to Kelvin

Standard gas calculations require temperature in Kelvin (K). Convert the given temperature from Celsius to Kelvin using the formula: $$T(K)=T(°C)+273.15$$ Thus, $$T=20+273.15=293.15\text{ K}$$

Calculate Number of Ozone Molecules Using Ideal Gas Law

To find the number of ozone molecules per liter, use the ideal gas law, $$PV=nRT$$ where $n$ is the number of moles. Solving for $n$: $$n=\frac{P_{\text{ozone}}\cdot V}{RT}$$ And for the number of molecules, $N$, use: $$N=n\cdot N_A$$ where $N_A\approx 6.022\times {10}^{23}\text{ molecules/mol}$. Here, assuming $V=1\text{ L}=0.001{\text{ m}}^3$ (for liter to m³ conversion and using $R=0.0821\text{ atm⋅L/mol⋅K}$): $$n\approx \frac{4.13\times {10}^{-7}\times 1}{0.0821\times 293.15}\approx 1.71\times {10}^{-8}$$ Then, calculate the number of molecules: $$N\approx 1.71\times {10}^{-8}\times 6.022\times {10}^{23}\approx 1.03\times {10}^{16}\text{ molecules/L}$$

Key concepts

Ozone Concentration

Ozone concentration in the air, especially in urban areas, can be a major concern due to its impact on health and the environment. Concentration is often denoted in parts per million (ppm), which indicates how much ozone is present relative to a million parts of air. A concentration of 0.42 ppm means that out of a million air molecules, 0.42 would be ozone molecules. This measurement helps assess air quality.

To use ozone concentration in calculations like determining partial pressure and the number of molecules, it's important to understand that this value represents a very tiny fraction of the atmosphere's volume since ppm involves very small numbers.

Partial Pressure

Partial pressure refers to the pressure that a single type of gas in a mixture of gases would exert if it occupied the entire volume by itself at the same temperature. When we're dealing with a mixture such as air, which contains many different gases, each component contributes part of the total pressure. This is the concept of partial pressure.

In the exercise, you calculated the partial pressure of ozone using the given concentration and total atmospheric pressure. The formula used is:

• Convert ppm to a fraction (0.42 ppm = 0.42/1,000,000).
• Multiply this fraction by the total atmospheric pressure (0.984 atm).

This method allows us to find the precise contribution of ozone to the air's total pressure, helping assess its effect and compliance with health guidelines.

Temperature Conversion

In scientific calculations, especially those involving the Ideal Gas Law, temperature must be in Kelvin. The Kelvin scale starts at absolute zero, making it ideal for calculations that rely on proportional relationships.

To convert a temperature from Celsius to Kelvin, which is necessary for using the Ideal Gas Law, you use the equation:

• $T(K)=T(°C)+273.15$

By converting the given 20.0°C to Kelvin ($T=293.15K$), you ensure you're using a consistent scale for temperature, avoiding errors in calculations related to gas properties.

Moles to Molecules Conversion

Moles and molecules are two ways of expressing the quantity of a substance, crucial in chemistry. A mole refers to Avogadro's number, which is approximately $6.022\times {10}^{23}$ entities (such as atoms or molecules). This makes conversion between moles and molecules straightforward as it's just a multiplication.

To find the number of molecules from moles, you use the equation:

• $N=n\times N_A$

where $N_A$ is Avogadro's number. Utilizing the Ideal Gas Law, we find the number of moles, $n$, and multiply by $N_A$ to convert to the number of molecules. In the exercise, this calculation determined the total number of ozone molecules present in a liter of air, giving insight into the concentration's physical manifestation.