Question

If the pressure exerted by ozone, \(\mathrm{O}_{3}\), in the stratosphere is \(3.0 \times 10^{-3} \mathrm{~atm}\) and the temperature is \(250 \mathrm{~K}\), how many ozone molecules are in a liter?

Short answer

There are approximately \(8.79 \times 10^{19}\) ozone molecules in a liter.

Step-by-step solution

Identify the given values and constants

We are given: - Pressure, P = \(3.0 \times 10^{-3} atm\) - Temperature, T = 250 K - Volume, V = 1L = 0.001 m^3 (as 1 liter = 1000 cubic centimeters = 0.001 cubic meters) - Gas constant, R = 0.0821 atm L/mol K (using this value since pressure is given in atm)

Use the Ideal Gas Law to calculate the amount of substance (n)

We will use the Ideal Gas Law equation, PV = nRT, and solve for 'n': \(n = \frac{PV}{RT}\) Substitute the given values: \(n = \frac{(3.0 \times 10^{-3} atm) (1L)}{(0.0821 \frac{atm*L}{mol*K})(250K)}\)

Perform calculations to find 'n'

Now, carry out the calculations: \(n = \frac{3.0 \times 10^{-3}}{(0.0821)(250)}\) \(n \approx 1.46 \times 10^{-4} mol\)

Convert moles to molecules using Avogadro's number

Now that we have the number of moles, we can convert it to molecules using Avogadro's number, which states that there are approximately \(6.022 \times 10^{23}\) molecules in one mole of a substance. Number of ozone molecules = \(n \times Avogadro's number\) = \((1.46 \times 10^{-4} mol) \times (6.022 \times 10^{23} molecules/mol)\)

Calculate the final value and present the result

Multiply the values: Number of ozone molecules \(\approx (1.46 \times 10^{-4}) \times (6.022 \times 10^{23})\) Number of ozone molecules \(\approx 8.79 \times 10^{19}\) Hence, there are approximately \(\boldsymbol{8.79 \times 10^{19}}\) ozone molecules in a liter.

Key concepts

Understanding Moles Conversion

The concept of moles conversion is essential in chemistry as it helps us quantify substances on a molecular level. A mole is a unit that represents a specific number of particles, such as atoms or molecules, similar to how a dozen represents twelve items. It's a bridge between the microscopic world of atoms and the macroscopic world we can measure.
When solving problems involving gases, we often use the Ideal Gas Law formula: \[ n = \frac{PV}{RT} \]where

• \(n\) is the number of moles,
• \(P\) is the pressure,
• \(V\) is the volume,
• and \(T\) is the temperature.

This equation helps us determine how many moles of a gas are present under specific conditions of pressure, volume, and temperature. Once we have calculated the moles, we can convert these to molecules using Avogadro's number.

The Role of Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, valued at approximately \(6.022 \times 10^{23}\) molecules per mole. This massive number allows chemists to count atoms, molecules, and other particles in a sample. By multiplying the number of moles by Avogadro's number, we find the total number of molecules present.
For example, in our exercise, we found approximately \(1.46 \times 10^{-4}\) moles of ozone in one liter of stratosphere air. To find out how many ozone molecules this represents, we calculated:\[1.46 \times 10^{-4} \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole} \]This calculation results in around \(8.79 \times 10^{19}\) molecules. Understanding Avogadro's number helps us comprehend the vast scale of atoms and molecules that make up the substances we encounter daily.

Why Stratospheric Pressure Matters

Stratospheric pressure refers to the pressure exerted by gases in the stratosphere, a layer of Earth's atmosphere. Although stratospheric pressure is much lower than at Earth's surface, it plays a key role in atmospheric science and chemistry.
The pressure affects how gases behave and interact. In our example, ozone exerts a pressure of \(3.0 \times 10^{-3} \text{ atm}\). Lower pressures like this one are typical in high-altitude layers such as the stratosphere, caused by fewer gas molecules being present compared to at sea level.
Understanding stratospheric conditions is crucial for studying phenomena like the ozone layer's role in blocking harmful UV radiation and how human activities impact atmospheric chemistry. By analyzing stratospheric pressures, scientists predict changes in weather patterns and assess environmental health.