The ozone layer is about 20 km thick, has an average total pressure of 10 mm Hg (1.3 ×10^-2 atm)

The ozone layer is about 20 km thick, has an average total...

The ozone layer is about 20 $\mathrm{km}$ thick, has an average total pressure of 10 $\mathrm{mm}$ Hg $\left(1.3 \times 10^{-2} \mathrm{atm}\right),$ and has an average temperature of 230 $\mathrm{K}$ . The partial pressure of ozone in the layer is only about $1.2 \times 10^{-6} \mathrm{mm}$ Hg $\left(1.6 \times 10^{-9} \mathrm{atm}\right) .$ How many meters thick would the layer be if all the ozone contained in it were compressed into a thin layer of pure $\mathrm{O}_{3}$ at STP?

Key Concepts

Conservation of Moles

When a gas is compressed or expanded under changing conditions, the total number of moles remains constant if there is no addition or removal of the gas. This concept is fundamental in problems where a gas is transferred between different states, such as from atmospheric conditions to STP, to help determine resulting physical dimensions like layer thickness.

Partial Pressure

Partial pressure is the pressure contribution of a single gas in a mixture of gases. It is an important concept because the behavior of each gas in a mixture often follows the ideal gas law independently. This concept is key in atmospheric science and chemical equilibrium problems, where the properties and reactions of individual gas components are analyzed.

Molar Volume

Molar volume is the volume occupied by one mole of a gas at a given temperature and pressure. At STP, this value is approximately 22.4 liters for an ideal gas. It provides a direct link between the number of moles of a gas and its volume, which is essential for calculations involving gas compression and expansion.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, number of moles, and temperature of a gas. It is fundamental for understanding how a gas behaves when it is compressed or expanded under different conditions. In problems involving changes in pressure, temperature, and volume, the ideal gas law provides the necessary framework to calculate one variable when the others are known.

Standard Temperature and Pressure (STP)

STP is a reference condition for gas calculations, usually defined as 0°C (273.15 K) and 1 atm (or 760 mm Hg) of pressure. It is used to simplify gas volume measurements, with one mole of an ideal gas occupying a standard molar volume (approximately 22.4 liters). Understanding STP is crucial when comparing gas behaviors or calculating densities and volumes under defined conditions.

Transcript

00:01 This is a great problem to understand the behavior of gases because this allows us to investigate a real -world application like the ozone layer.

00:11 So this is a very quick visualization of the information in the problem.

00:16 We have this ozone layer and we know that this ozone layer is 20 kilometers thick.

00:23 We're also told that our temperature is 230 kelvin.

00:27 We know that our total pressure is 1 .3 times 10 to the negative.

00:31 Of second atm.

00:34 And we're going to say let lambda be area.

00:37 If lambda is confusing, you just think of it as a variable.

00:41 It's nothing different.

00:43 We're going to say let lambda be area.

00:46 So we can say that our volume will be equivalent to 20 times lambda.

00:50 Remember, lambda is area.

00:53 We should know this equation where the p total, pressure total, times v total, our volume, is equivalent to our n total, the number of moles times rt.

01:06 Now we can plug things in that we know to this equation.

01:10 Define n total, n total or our number of moles, will be equivalent to 1 .3 times 10 to the negative second times 20 lambda divided by 0 .0821 times 230.

01:25 This will simplify, so our number of total moles is 0 .0136 times lambda.

01:35 Now what we can do is we can find the mole fraction of ozone, which is 03.

01:40 The mole fraction of ozone will be equivalent to the moles of ozone divided by our total number of moles.

01:49 And if we rearrange this equation or make a substitution, we should know that the moles or pardon me, the mole fraction of ozone is equivalent to the pressure of ozone divided by the total pressure...

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