Problem 30 - General Chemistry: Principles and Modern Applications

Question

A typical concentration of \(\mathrm{O}_{3}\) in the ozone layer is \(5 \times 10^{12} \mathrm{O}_{3}\) molecules \(\mathrm{cm}^{-3} .\) What is the partial pressure of \(\mathrm{O}_{3},\) expressed in millimeters of mercury, in that layer? Assume a temperature of \(220 \mathrm{K}\).

Step-by-Step Solution

Answer

The partial pressure of \(O_{3}\) in the ozone layer, at a temperature of 220K, is \(1.1 \times 10^{-3}\) mmHg.

Step 1: Conversion to moles

First, we need to convert \(5 \times 10^{12} O_{3}\) molecules/cm\(^3\) to moles/cm\(^3\). Given that 1 mole of any substance has \(6.022 \times 10^{23}\) molecules (Avogadro's Number), this can be done using the following equation: Mole concentration = Molecule concentration / Avogadro's Number = \(5 \times 10^{12} O_{3}\) molecules/cm\(^3\) / \(6.022 \times 10^{23}\) molecules/mole = \(8.3 \times 10^{-12}\) mole/cm\(^3\)

Step 2: Apply ideal gas law

The ideal gas law (PV = nRT) can then be used to calculate pressure. Rearranging for the pressure gives us P=nRT/V. Note that the gas constant R is in L atm/mol K and the given temperature T is in Kelvin, so the pressure P will be in atm. Subtituting the values we have P = \(8.3 \times 10^{-12}\) mole/cm\(^3\) * 0.0821 L atm/mol K * 220 K. But, 1 cm\(^3\) equals 1e-3 L, so the volume V needs to be adjusted accordingly. Therefore, P= \(1.5 \times 10^{-6}\) atm.

Step 3: Conversion to mmHg

Lastly, the pressure needs to be converted to mmHg, which can be done using the conversion 1 atm = 760 mmHg. Thus, P = \(1.5 \times 10^{-6}\) atm * 760 mmHg/atm = 1.1 \times 10^{-3} mmHg.

Key Concepts

Avogadro's Number

Avogadro's number, a fundamental concept in chemistry, is pivotal for understanding the relationship between molecules and moles. It is defined as the number of constituent particles (typically atoms or molecules) found in one mole of a substance. This number is approximately \(6.022 \times 10^{23}\), a constant that allows scientists to convert between the observable bulk quantities of materials and the number of atoms or molecules they consist of.

This conversion is essential in chemical calculations, such as when figuring out the number of molecules present in a given concentration. For instance, if we have \(5 \times 10^{12}\) ozone molecules per cubic centimeter, we divide by Avogadro's number to convert this to moles per cubic centimeter, yielding \(8.3 \times 10^{-12}\) mole/cm\(^3\).

Understanding how to use Avogadro's number is critical for performing accurate chemical calculations, as it provides a bridge between the macroscopic measurements (we can see and measure) and the microscopic world of atoms and molecules(we cannot see).

Mole Conversion

Mole conversion is a crucial process in chemical problem-solving, helping bridge the gap between molecular-scale reactions and macroscopic quantities. It involves converting a given amount from molecules or atoms to moles, or vice versa.

In this context, converting the concentration of ozone molecules to moles is straightforward thanks to Avogadro's number. Given that the concentration of ozone in the ozone layer is \(5 \times 10^{12}\) molecules/cm\(^3\), we use the formula:

• \(\text{Mole concentration} = \text{Molecule concentration} / \text{Avogadro's Number}\)

Performing the calculation, we find:

• \(8.3 \times 10^{-12}\) mole/cm\(^3\)

Understanding mole conversions is fundamental for effectively using the ideal gas law and other chemical equations, as they often require input values in moles.

Pressure Conversion

Pressure conversion is an essential skill in chemistry, as pressures are often expressed in different units. Understanding how to convert between these units is vital for interpreting chemical equations and results correctly.

For example, in the context of the ozone problem, we initially determined the partial pressure in atmospheres using the ideal gas law. However, we needed the final answer in millimeters of mercury (mmHg). The conversion is fairly straightforward:

• 1 atmosphere (atm) is equal to 760 mmHg.

This allows us to convert \(1.5 \times 10^{-6}\) atm to mmHg by multiplying by 760. This results in a partial ozone pressure of \(1.1 \times 10^{-3}\) mmHg.

Mastering pressure conversions ensures that you can deal with a variety of experimental and theoretical situations seamlessly.

Ozone Layer Concentration

The concentration of ozone in the ozone layer is important for understanding its role in protecting Earth from harmful ultraviolet (UV) radiation. This concentration is typically measured in terms of particles per unit volume, often molecules per cubic centimeter.

In the problem given, the concentration is \(5 \times 10^{12}\) molecules/cm\(^3\). To work with this concentration practically, scientists often convert it to moles, which aligns with the calculations needed to determine properties like pressure using the ideal gas law.

By doing these conversions, we can derive crucial environmental insights, such as the extent of ozone's protective capacity at different altitudes and under different atmospheric conditions.

Understanding ozone concentrations and how to convert these into usable chemical units is key to studying atmospheric science and environmental chemistry.