### 2.1 Gas Laws

Understanding atmospheric thermodynamics begins with the gas laws that you learned in chemistry. Because these laws are so important, we will review them again here and put them in forms that are particularly useful for atmospheric science. **You will want to memorize these laws** because they will be used again and again in many other areas of atmospheric science, including cloud physics, atmospheric structure, dynamics, radiation, boundary layer, and even forecasting.

#### Looking Ahead

Before you begin this lesson's reading, I would like to remind you of the discussion activity for this lesson. This week's discussion activity will ask you to take what you learn throughout the lesson to answer an atmospheric problem. You will not need to post your discussion response until you have read the whole lesson, but keep the question in mind as you read:

**This week's topic is a hypothetical question involving stability. The troposphere always has a capping temperature inversion—it's called the stratosphere. The tropopause is about 16 km high in the tropics and lowers to about 10 km at high latitudes. The stratosphere exists because solar ultraviolet light makes ozone and then a few percent of the solar radiation is absorbed by stratospheric ozone, heating the air and causing the inversion. Suppose that there was no ozone layer and hence no stratosphere caused by solar UV heating of ozone.**

**Would storms in the troposphere be different if there was no stratosphere to act like a capping inversion? And if so, how?**

You will use what you have learned in this lesson about the atmosphere's pressure structure and stability to help you to think about this problem and to formulate your answer and discussions. So, think about this question as you read through the lesson. You'll have a chance to submit your response in 2.6!

### Ideal Gas Law

The atmosphere is a mixture of gases that can be compressed or expanded in a way that obeys the Ideal Gas Law:

$$pV=N{R}^{*}T$$

where *p* is pressure (Pa = kg m^{–1} s^{–2}), *V* is the volume (m^{3}), *N* is the number of moles, *R** is the gas constant (8.314 J K^{–1} mole^{–1}), and *T* is the temperature (K). Note also that both sides of the Ideal Gas Law equation have the dimension of energy (J = kg m^{2} s^{–2}).

Recall that a mole is 6.02 x 10^{23} molecules (Avogadro’s Number). Equation 2.1 is a form of the ideal gas law that is independent of the type of molecule or mixture of molecules. A mole is a mole no matter its type. The video below (6:17) provides a brief review of the Ideal Gas Law. Note that the notation in the video differs slightly from our notation by using *n* for *N*, *P* for *p*, and *R* for *R**.

So here I have a tank filled with gas. And these little dots represent some of the gas particles that would be in this tank. The arrows I put in here because all of these particles are in constant random motion. They're like a bunch of hyperactive little kids, running into each other all the time, banging into the sides of the container, and so forth. So we've got this tank of gas. Let's think about the characteristics that we could use to describe it. So one of the things that we could do is we could say what its temperature is. The higher the temperature, remember, the faster these gas particles are moving around, so temperature is very important when we talk about gas. Temperature for gases should always be reported in Kelvin. So we could say, for example, that the temperature of this guy here is 313 Kelvin. That's how hot these gas particles in the sample are. When you talk about gas, another important characteristic is pressure. How hard are these gas particles bouncing against the sides of the tank? How much pressure are they exerting on them? And we could measure these with a pressure gauge or something like that on the top of this tank. We could say, the pressure for this is 3.18 atm. That might be a pressure. And another thing that we spend a lot of time talking about when it comes to gas is volume. And again, I have these letters here that are how each one of these things are abbreviated. Volume, V, volume of this tank might be something like 95.2 liters. And finally, look at these particles that I've drawn. There is a certain amount of gas that's in here. And the amount of gas, which is abbreviated by the little letter n, is usually reported in moles, which is a convenient measure of how much of something we have. So we could say that the amount of gas in this tank is 7.5 moles. Now, whenever we have a sample of gas like this, if it's a tank or it's in a balloon or wherever it is, we can describe-- we can give it these various characteristics. And it turns out that also, for any sample of gas, if we know three of these characteristics, we can figure out what the fourth is. All we need to do is know three. And in order to do that, we use an equation that's a representation of the Ideal Gas Law. And it's written as P times V, pressure times volume, equals n, the amount of gas, times R times T, temperature. I'll get to R in a second. Don't worry about it for right now. It's going to be a number that we know. So let's say, for example, that we didn't know what pressure was, but we still knew the temperature, volume, and the amount of gas. No big deal. We could take the equation, PV equals nRT, and rearrange it. Divide both sides by V. Get rid of the V. And then we'd have P equals nRT divided by V. Plug these values in, and we could figure out what the pressure was. Or let's say that we knew what the pressure was of a particular gas sample. We know what the temperature was in a volume. But we didn't know what the amount of gas was. We don't know how much we had. We could figure out that fourth characteristic by rearranging the Ideal Gas Law for n, canceling out R and T on one side, rearranging it to solve for n. And then we could plug in the pressure, the volume, and the temperature, and we could figure out the amount of gas. So in other words, if we know three of these characteristics, we can always figure out what the fourth is. So you may be asking yourself, so R-- what's R? R is what we call a constant. It's a number that we know ahead of time that doesn't depend on the variables in our problem. The R that I'm going to be using most of the time for the videos is 0.0821 liters times atm divided by Kelvin times moles. Now notice that this is a fraction. It has both a top and a bottom. And it also is not just a number, but it has units. And check this out-- the units on R match the units in my problem. They match the characteristics that I'd be using. So I have liters here, liters here, atm, atm, Kelvin, Kelvin, and moles, moles. You always want the units on R to match the units of the characteristics in your Ideal Gas Problem. So because you always want the units to match, there are also different values of R, although I'm going to be using this mostly for the videos I'm doing. For example, let's say that instead of atm, I was using a pressure that was in millimeters of mercury. In this case, I wouldn't want to use this R here. I'd want to use this R here, so that the units match-- millimeters of mercury here, millimeters of mercury here, and the number's different-- 62.4. So again, that's what I use here. Let's say that instead of millimeters of mercury, my pressure was given to me in kPa. I would then use this value of R so that the units match. I've got kPa here, kPa here, and all the others are the same, so 8.31 for that. Now as I keep saying, in most of the videos that I'm going to be doing, I'm going to be using this top R with atm. But you may be asked by your teacher to use a different R. It's no big deal. That's probably just because they're giving you problems that have different pressure units, and they want the pressure units to match. So don't worry at all if you're using one of these other R's. Setting up and solving the Ideal Gas Law is exactly the same. No matter which of these R's you use, it's just a matter of plugging a different R in at the very end. So no matter which one you're using, you should be able to follow all these lessons, and it should all make sense.

Usually in the atmosphere we do not know the exact volume of an air parcel or air mass. To solve this problem, we can rewrite the Ideal Gas Law in a different useful form if we divide *N* by *V* and then multiply by the average mass per mole of air to get the **mass density**:

$$\rho =\frac{NM}{V}$$

where *M* is the molar mass (kg mol^{–1}). Density has SI units of kg m^{–3}. The Greek symbol *ρ* (rho) is used for density and should not be confused with the symbol for pressure, *p*.

Thus we can put density in the Ideal Gas Law:

$$p=\frac{\rho {R}^{*}T}{M}or\rho =\frac{Mp}{{R}^{*}T}$$

Density is an incredibly important quantity in meteorology. Air that is more dense than its surroundings (often called its environment) sinks, while air that is less dense than its surroundings rises. Note that density depends on temperature, pressure, and the average molar mass of the air parcel. The average molar mass depends on the atmospheric composition and is just the sum of the mole fraction of each type of molecule times the molar mass of each molecular constituent:

$${M}_{average}=\frac{{\displaystyle \sum _{i}{N}_{i}{M}_{i}}}{{\displaystyle \sum _{i}{N}_{i}}}=\frac{{\displaystyle \sum _{i}{N}_{i}{M}_{i}}}{N}={\displaystyle \sum _{i}\frac{{N}_{i}}{N}{M}_{i}}={\displaystyle \sum _{i}{f}_{i}{M}_{i}}$$

where the* i* subscript represents atmospheric components, *N* is the number of moles,* M* is the molar mass, and *f* is the mole fraction. This video (3:19) shows you how to find the gas density using the Ideal Gas Law. You will note that the person uses pressure in kPa and molar mass in g/mol. Since kPa = 1000 Pa and g = 1/1000 kg, the two factors of 1000 cancel when he multiplies them together and he can get away with using these units. I recommend always converting to SI units to avoid confusion. Also, note that the symbol for density used in the video is *d*, which is different from what we have used (*ρ*, the convention in atmospheric science).

Hey guys how do you solve ideal gas law questions involving density? The key is to have a formula or know how to derive the formula on your own. Remember density is mass over volume. Now, the way that mass is found in the ideal gas law equation is in n because the number of moles is the same as mass over molar mass. So, check this set. I'm going to replace n with mass over molar mass, and then I'm going to rearrange for m over v. I'm going to undo division by molar mass on the other side and then I'm going to undo multiplication by RT and bring my V over. Here's a what I mean. P times the molar mass divided by RT gives me mass over volume. Mass over volume is density and so my equation is density equals pressure times molar mass divided by RT. We can now use this equation to find the density of oxygen at 55 Celsius and a hundred and three kilopascals. So let's do it. The density is pressure that's 103 kilopascals times molar mass for oxygen. That's 32 grams per mole. R, now, I'm going to put my volume in liters and I'm going to put my pressure in kilopascals which means the relevant are that I want is 8.314 liters, kilopascals per mole Kelvin and my temperature in Kelvin is the temperature in Celsius plus 273, which gives me 328 Kelvin. And all of these units should cancel out to give me a density unit. Kelvin cancels of Kelvin per moles cancel / moles kilopascals canceled the scale pascals and left with grams per liter. Let's do this on the calculator 103 times 32 divided 8.314 divided 328. That's 1.21 grams per liter. That may not seem like a lot, but remember you're dealing with the gas here. If you have a 1-liter balloon how much is it actually going to weigh? Probably the amount of the rubber plus like a gram or so. This here is the density of oxygen gas at 55 and 103 kilopascals. This is your density formula in terms of the ideal gas law. Be able to use it. Best of luck!

Solve the following problem on your own. After arriving at your own answer, click on the link to check your work.

#### Example

Let’s calculate the density of dry air where you live. We will use the Ideal Gas Law and account for the three most abundant gases in the atmosphere: nitrogen, oxygen, and argon. The mole fractions of the gases are 0.78, 0.21, and 0.01, respectively. *M* is the molar mass of air; *M* = 0.029 kg mol^{–1}, which is just an average that accounts for the mole fractions of the three gases:

*R** = 8.314 J K^{–1} mol^{–1}. Here, *p* = 960 hPa = 9.6 x 10^{4} Pa and *T* = 20 ^{o}C = 293 K.

**Click for answer.**

Putting these values into the equation 2.3, we get that the dry air density is 1.1 kg m^{–3}.

What would the density be if the room were filled with helium and not dry air at the same pressure and temperature?

**Click for answer.**

Helium density = (pressure times molar mass of helium)/(Ideal Gas Law constant in SI units times temperature in K) = 0.16 kg m^{–3}

#### Dry Air

Often in meteorology we use mass-specific gas laws so that we must specify the gas that we are talking about, usually only dry air (N_{2} + O_{2} + Ar + CO_{2} +…) or water vapor (gaseous H_{2}O). We can divide *R ^{*}* by

*M*to get a mass-specific gas constant, such as

_{i}*R*.

_{d}= R*/M_{dry air}Thus, we will use the following form of the Ideal Gas Law **for dry air**:

$${p}_{d}={\rho}_{d}{R}_{d}T$$

where:

${R}_{d}=\frac{R*}{{M}_{dryair}}=\frac{8.314{\text{JK}}^{-1}{\text{mol}}^{-1}}{0.02897{\text{kgmol}}^{-1}}=287{\text{m}}^{2}{\text{s}}^{-2}{\text{K}}^{-1}=287{\text{Jkg}}^{-1}{\text{K}}^{-1}$ .

*M _{dry air}* is 0.02897 kg mol

^{–1}, which is the average of the molar masses of the gases in a dry atmosphere computed to four significant figures.

Note that *p* must be in Pascals (Pa), which is 1/100th of a mb (a.k.a, hPa), and *T* must be in Kelvin (K).

#### Water Vapor

We can do the same procedure **for water vapor**:

$${p}_{v}\equiv e={\rho}_{v}{R}_{v}T$$

where ${R}_{v}=\frac{R*}{{M}_{watervapor}}=\frac{8.314{\text{JK}}^{-1}{\text{mol}}^{-1}}{0.01802{\text{kgmol}}^{-1}}=461{\text{m}}^{2}{\text{s}}^{-2}{\text{K}}^{-1}=461{\text{Jkg}}^{-1}{\text{K}}^{-1}$ .

Typically *e* is used to denote the water vapor pressure, which is also called the water vapor partial pressure.

### Dalton’s Law

This gas law is used often in meteorology. Applied to the atmosphere, it says that the total pressure is the sum of the partial pressures for dry air and water vapor:

$$p={p}_{d}+{p}_{H2O}={p}_{d}+e$$

Imagine that we put moist air and an absorbent in a jar and screw the lid on the jar. If we keep the temperature constant as the absorbent pulls water vapor out of the air, the pressure inside the jar will drop to *p _{d}*. Always keep in mind that when we measure pressure in the atmosphere, we are measuring the

**pressure, which includes the partial pressures of dry air and water vapor.**

*total*So it follows that the **density of dry air and water vapor **also add:

$$\rho ={\rho}_{d}+{\rho}_{v}$$

#### Check Your Understanding

Suppose we have two air parcels that are the same size and have the same pressure and temperature, but one is dry and the other is moist air. Which one is less dense?

**Click for answer.**

We can solve this one without knowing the pressure, temperature, or volume. Let’s assume that 98% of the molecules are dry air, which means the remaining 2% are dry air in the first case and water vapor in second case. Dry air is 0.029 kg mol^{–1} and water vapor is 0.018 kg mol^{–1}, so 2% of the moist air is lighter than the 2% of dry air, and when we consider the total air, this means that *for the same temperature and pressure, moist air is always less dense than dry air*.

### Virtual Temperature

Suppose there are two air parcels with different temperatures and water vapor amounts but the same pressure. Which one has a lower density? We can calculate the density to determine which one is lighter, but there is another way to do this comparison. Virtual temperature, *T _{v}*, is defined as the temperature dry air must have so that its density equals that of ambient moist air. Thus, virtual temperature is a property of the ambient moist air. Because the air density depends on the amount of moisture (for the same pressure and temperature), we have a hard time determining if the air parcel is more or less dense relative to its surroundings, which may have a different temperature and amount of water vapor. It is useful to pretend that the moist parcel is a dry parcel and to account for the difference in density by determining the temperature that the dry parcel would need to have in order to have the same density as the moist air parcel.

We can define the amount of moisture in the air by a quantity called **specific humidity, q**:

$$q=\frac{{\rho}_{v}}{{\rho}_{d}+{\rho}_{v}}$$

We see that *q* is just the fraction of water vapor density relative to the total moist air density. Usually *q* is given in units of g of water vapor per kg of dry air, or g kg^{–1}.

Using the Ideal Gas Law and Dalton’s Law, we can derive the **equation for virtual temperature**:

$${T}_{v}=T\left[1+0.61q\right]$$

where *T* and *T _{v}* have units of Kelvin (not

^{o}C and certainly not

^{o}F!) and

*q*must be unitless (e.g., kg kg

^{–1}).

Note that moist air always has a higher virtual temperature than dry air that has the same temperature as the moist air because, as noted above, moist air is always less dense than dry air for the same temperature and pressure. Note also that for dry air, *q* = 0 and the virtual temperature is the same as the temperature.

Solve the following problem on your own. After arriving at your own answer, click on the link to check your work.

#### Check Your Understanding

Consider a blob of air (*T _{blob}* = 25

^{ o}C,

*q*= 10 g kg

_{blob}^{–1}) at the same pressure level as a surrounding environment (

*T*= 26

_{env}^{ o}C and

*q*

_{env}= 1 g kg

^{–1}). If the blob has a lower density than its environment, then it will rise. Does it rise?

**Click for answer.**

We will use equation 2.10. Remember to convert *T* from ^{o}C to K and *q* from g kg^{–1} to kg kg^{–1}!

$$\begin{array}{l}{T}_{vblob}=(25+273)\left[1+0.61\cdot .010\right]=299.8K=26.8{}^{o}C\\ {T}_{venv}=(26+273)\left[1+0.61\cdot .001\right]=299.2K=26.2{}^{o}C\end{array}$$

We see that the blob is less dense than its environment and so will rise. This difference of 0.6 ^{o}C may seem small, but it makes a huge difference in upward motion.

The following are some mistakes that are commonly made in the above calculations:

- not converting from
^{o}C to K:

$$\begin{array}{l}{T}_{vblob}=(25)\left[1+0.61\cdot .010\right]=25.15{}^{o}C\\ {T}_{venv}=(26)\left[1+0.61\cdot .001\right]=26.02{}^{o}C\end{array}$$

We calculate that *T _{vblob} < T_{venv}*, which is the wrong answer.

- not converting
*q*from g/kg to kg/kg:

$$\begin{array}{l}{T}_{vblob}=(25+273)\left[1+0.61\cdot 10\right]=2115K=1842{}^{o}C\\ {T}_{venv}=(26+273)\left[1+0.61\cdot 1\right]=481K=208{}^{o}C\end{array}$$

We calculate that *T _{vblob} > T_{venv}*, which is correct in this case, but the numbers are crazy! After you complete your calculations, if the numbers you get just don’t seem right—like these—then you know that you have made a mistake in the calculation. Go looking for the mistake. Don’t submit an answer that makes no sense.

Once we find *T _{v}*, we can easily find the density of a moist parcel by using equation 2.5, in which we substitute

*T*for

_{v}*T*. Thus,

$${\rho}_{}=\frac{{p}_{}}{{R}_{d}{T}_{v}}$$

#### Quiz 2-1: What will that air parcel do?

This quiz will give you practice calculating the virtual temperature and density using the Excel workbook that you set up in the last lesson.

- Go to Canvas and find
**Practice Quiz 2-1**. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz.*I strongly suggest that you enter the equations for density and for virtual temperature in your Excel worksheet and use them to do all your calculations of density and virtual temperature on both the practice quiz and the quiz.* - When you feel you are ready, take
**Quiz 2-1**. You will be allowed to take this quiz only**once**. This quiz is timed, so after you start, you will have a limited amount of time to complete it and submit it. Good luck!