## MCAT General Chemistry Review

__Part I ____Review__

__Part I__

__Review__

__Chapter 7: The Gas Phase__

__Chapter 7: The Gas Phase__

We are literally surrounded by gas. We walk through it, run through it, breathe it in, breathe it out, burp it, pass it, and if it weren’t for the fact that we are denser than air, we’d be swimming in it. Gases behave in ways that we find useful, interesting, and entertaining. How else to explain the almost universal human delight in balloons: helium balloons, hot air balloons, circus balloon animals, *99 Luftballons*? The topic of discussion for this chapter is not, alas, the art of fashioning balloon animals or the deeper meaning of the Cold War–era German protest song. We will discuss the MCAT favorites—the gas phase and the ideal gas laws. We will begin our discussion of ideal gases and the __kinetic molecular theory__ that describes them. We will then examine each of the laws that govern the behavior of ideal gases and conclude with an evaluation of the ways in which the behavior of real gases deviate from that predicted by the ideal gas law.

Before we get started with our discussion of gases, here’s a fun little experiment that you can try on your own, not only to get you thinking about the characteristics of gases but also to help you connect gas behaviors to some of the principles of Newtonian physics that you must know and understand for Test Day. (The MCAT will be making these kinds of connections, so you should be preparing for that now.)

The next time you’re leaving a birthday party, holiday celebration, bar or bat mitzvah, christening, wedding, or funeral, snag a helium balloon on your way out. (Balloons at funerals? Hey—some people like to go out smiling!) Tie the helium balloon to the gearshift lever between the front seats, making sure that the balloon is floating freely. Once you’re on an open road, accelerate the car abruptly, and as you do, watch the balloon. What do you predict will happen to the balloon as the car accelerates? What do you observe? You might think, based on how you feel when you are sitting in a car accelerating forward , that the balloon will be pushed toward the back of the car due to its inertia. After all, isn’t this what happens to you? You feel pushed back into your seat as a result of your inertia, which is resisting the car’s forward force and acceleration. However, the balloon’s movement isn’t what we might predict. In fact, it’s the opposite: The balloon shifts forward as the car accelerates forward because the balloon is filled with helium, one of the noble gases. The molecular weight of helium is 4 grams/mole, while that of air, which is mostly nitrogen and oxygen, is about 29 grams/mole. This means that air is about seven times denser than helium. Because the denser air in which the balloon is floating has more mass than the helium-filled balloon, the air will have greater inertia. In fact, we can approximate the balloon’s inertia as practically nonexistent. Therefore, as the car accelerates forward, everything that has significant mass, including the air in the car, resists the forward motion (has inertia) and shifts toward the back of the car (even though, of course, everything in the car is accelerating forward, just not as quickly as the car itself ). As the air shifts toward the back, a pressure gradient builds up such that there is greater air pressure in the back of the car than in the front, and this pressure difference results in a pushing force against the balloon that is directed from the back toward the front. Responding to this force, the balloon shifts forward in the direction of the car’s acceleration. Who would think that general chemistry and physics could be so much fun? Well, if you’ve been paying attention: We do!

**The Gas Phase**

Matter can exist in three different physical forms, called **phases** or **states**: gas, liquid, and solid. We will discuss liquids and solids in Chapter 8, Phases and Phase Changes. The **gaseous phase** is the simplest to understand, since all gases display similar behavior and follow similar laws regardless of their particular chemical identities. Like liquids, gases are classified as fluids because they can flow. The atoms or molecules in a gaseous sample move rapidly and are far apart from each other. In addition, only very weak intermolecular forces exist between gas particles; this results in certain characteristic physical properties, such as the ability to expand to fill any volume and to take on the shape of a container. (This last characteristic defines fluids—liquids and gases—generally.) Gases are also easily, although not infinitely, compressible, which distinguishes them from liquids.

We can define the state of a gaseous sample generally by four variables: pressure (*P*), volume (*V* ), temperature (*T* ), and number of moles (*n*). Gas pressures are usually expressed in units of atmospheres (atm) or in units of millimeters of mercury (mm Hg), which are equivalent to __torr__. The SI unit for pressure, however, is the pascal (Pa). The mathematical relationships among all of these units are

1 atm = 760 mm Hg = 760 torr = 101.325 kPa

On the MCAT, you’ll encounter any or all of these units, so become familiar with them through your practice problems. The volume of a gas is generally expressed in liters (L) or milliliters (mL). Temperature is usually given in __Kelvin (K)__. Many processes involving gases take place under certain conditions, called **standard temperature and pressure**, or **STP**, which refers to conditions of 273.13 K (0°C) and 1 atm.

Please carefully note that **STP conditions** are not identical to **standard state conditions.** The two standards involve different temperatures and are used for different purposes. STP (273 K and 1 atm) is generally used for gas law calculations; standard state conditions (298 K and 1 atm) are used when measuring standard enthalpy, entropy, free energy changes, and voltage.

**MCAT Expertise**

On the MCAT, remember that STP is different from standard state. Temperature at STP is 0°C (273.15 K). Temperature at standard state is 25°C (298.15 K).

**Ideal Gases**

When we examine the behavior of gases under varying conditions of temperature and pressure, we assume that the gases are ideal. An ideal gas represents a hypothetical gas whose molecules have no intermolecular forces and occupy no volume. Although real gases deviate from this ideal behavior at high pressures and low temperatures, many real gases demonstrate behavior that is close to ideal.

**Kinetic Molecular Theory of Gases**

This mouthful of a theory was developed in the second half of the 19th century, well after the laws describing gas behavior had been developed. In fact, the **kinetic molecular theory** was developed to explain the behavior of gases, which the laws merely described. The gas laws demonstrate that all gases show similar physical characteristics and behavior irrespective of their particular chemical identity. The behavior of real gases deviates from the ideal behavior predicted under the assumptions of this theory, but these deviations may be corrected for in calculations. The combined efforts of Boltzmann, Maxwell, and others led to a simple explanation of gaseous molecular behavior based on the motion of individual molecules. Like the gas laws, which we will examine shortly, the kinetic molecular theory was developed in reference to ideal gases, although it can be applied with reasonable accuracy to real gases as well.

**ASSUMPTIONS OF THE KINETIC MOLECULAR THEORY**

1. Gases are made up of particles whose volumes are negligible compared to the container volume.

2. Gas atoms or molecules exhibit no intermolecular attractions or repulsions.

3. Gas particles are in continuous, random motion, undergoing collisions with other particles and the container walls.

4. Collisions between any two gas particles are elastic, meaning that there is conservation of both momentum and kinetic energy.

5. The average kinetic energy of gas particles is proportional to the absolute temperature (in Kelvin) of the gas, and it is the same for all gases at a given temperature, irrespective of chemical identity or atomic mass.

**APPLICATIONS OF THE KINETIC MOLECULAR THEORY OF GASES**

It’s fairly straightforward to imagine, based on the assumptions just listed, gas particles as lots of little rubber balls bouncing into and off each other and off the walls of their container. Of course, rubber balls, like __real gas__ particles, have measurable mass and volume, and not even the bounciest rubber balls will collide in a completely elastic manner.

**Average Molecular Speeds**

According to the kinetic molecular theory of gases, the average kinetic energy of a gas particle is proportional to the absolute temperature of the gas:

where k is the Boltzmann constant, which serves as a bridge between macroscopic and microscopic behavior (that is, as a bridge between the behavior of the gas as a whole and the individual gas molecules). This equation shows that the speed of a gas particle is related to its absolute temperature. However, because of the large number of rapidly and randomly moving gas particles, which may travel distances as short as 6 × 10^{-6} cm before colliding with another particle or the container wall, the speed of an individual gas molecule is nearly impossible to define. Therefore, the speeds of gases are defined in terms of their average molecular speed. One way to define an average speed is to determine the average kinetic energy per particle and then calculate the speed to which this corresponds. The resultant quantity, known as the root-mean-square speed (*u*_{rms}), is given by the following equation:

where R is ideal gas constant and *MM* is the molecular mass.

**MCAT Expertise**

Again, understanding concepts will be much more fruitful on Test Day than memorizing all of the facts. The higher the temperature, the faster the molecules move. The larger the molecules, they slower they move.

A __Maxwell-Boltzmann distribution__ curve shows the distribution of speeds of gas particles at a given temperature. __Figure 7.1__ shows a distribution curve of molecular speeds at two temperatures, *T*_{1} and *T*_{2}, where *T*_{2} is greater than *T*_{1}. Notice that the bell-shaped curve flattens and shifts to the right as the temperature increases, indicating that at higher temperatures, more molecules are moving at higher speeds.

__Figure 7.1__

**Example:** What is the average speed of sulfur dioxide molecules at 37°C?

**Solution:** The __gas constant R__ = 8.314 J/(K • mol) should be used, and *MM* must be expressed in kg/mol.

Use the conversion factor 1 J = 1 kg • m^{2}/s^{2}:

__Graham’s Law__** of **__Diffusion__** and Effusion**

Ours is a highly aromatic world. The smells and aromas of food, flowers, exhaust, rot, wet dog, and skunk are instantly recognizable and elicit in us reactions of delight or disgust. In our introduction to Chapter 4, Compounds and Stoichiometry, we explained that our sense of smell depends upon the interactions between vaporized compounds and the chemical receptors of the olfactory epithelium located inside our nasal cavities. The rate at which those gaseous compounds travel will determine what we’d smell first: a flower or a skunk.

The movement of gas molecules through a mixture (such as air) is called **diffusion**. The kinetic molecular theory of gases predicts that heavier gases diffuse more slowly than lighter ones do because of their differing average speeds. (Because all gas particles have the same average kinetic energy at the same temperature, it must be that particles with greater mass travel at a slower average velocity.) In 1832, Thomas Graham showed mathematically that under isothermal and isobaric conditions, the rates at which two gases diffuse are inversely proportional to the square root of their molar masses. Thus,

where *r*_{1} and *r*_{2} are the diffusion rates of gas 1 and gas 2, respectively, and *MM*_{1} and *MM*_{2} are the molar masses of gas 1 and gas 2, respectively. You know by now that the MCAT commonly tests students’ understanding of ratios. From this equation, you can see that a gas that has a molar mass four times that of another gas will travel half as fast as the lighter gas.

**Key Concept**

Diffusion: When gases mix with one another. Effusion: When a gas moves through a small hole under pressure. Both will be slower for larger molecules.

Effusion is the flow of gas particles under pressure from one compartment to another through a small opening. Graham used the kinetic molecular theory of gases to show that for two gases at the same temperature, the rates of effusion are proportional to the average speeds. He then expressed the rates of effusion in terms of molar mass and found that the relationship is the same as that for diffusion:

Latex balloons are often filled with a 60/40 mixture of helium and air. Latex is a fairly porous material that allows for the effusion of the gas mixture contained inside. Since the weighted average molar mass of air (consisting of about 78% N_{2} and 21% O_{2}) is about 29 grams/mole and the molar mass of helium is 4 grams/ mole, the helium gas will effuse almost three times faster than the air. This is why helium balloons have such a fleeting life span and perhaps explains in part their ability to enchant us.

**Ideal Gas Behavior**

The __ideal gas law__ was first stated in 1834 by Benoît Paul Émile Clapeyron, more than 170 years after Sir Robert Boyle has pesrformed his experimental studies on the relationship between pressure and volume in the gas state. In fact, by the time the ideal gas law found its expression, __Boyle’s law__,__Charles’ law__, and even __Dalton’s law__ had already been well established. Historical considerations aside, it will benefit us to examine the ideal gas law first so that we can then understand the other laws, which had been “discovered” first, to be only special cases of the ideal gas law.

One more important discovery that preceded Clapeyron’s formulation of the ideal gas law was Amedeo Avogadro’s formulation in 1811, known as __Avogadro’s principle__, that all gases at a constant temperature and pressure occupy volumes that are directly proportional to the number of moles of gas present. Equal amounts of all gases at the same temperature and pressure will occupy equal volumes. For example, one mole of any gas, irrespective of its chemical identity, will occupy 22.4 liters at STP.

where *n*_{1} and *n*_{2} are the number of moles of gas 1 and gas 2, respectively, and *V*_{1} and *V*_{2} are the volumes of the gases, respectively.

**IDEAL GAS LAW.**

The ideal gas law shows the relationships among four variables that define a sample of gas: pressure (*P*), volume (*V*), temperature (*T* ), and number of moles (*n*). The law combines the mathematical relationships earlier determined by the work of Boyle, Charles, and Gay-Lussac with Avogadro’s principle and is represented by this equation:

*PV* = *n*R*T*

where R is a constant known as the gas constant, which has a value of 8.21 × 10^{-2} (L•atm)/(mol•K). Be aware that the gas constant can be expressed in other units.

**MCAT Expertise**

*PV* = *n*R*T*. Knowing this equation means we can derive others based on the answer we are looking to find.

On the MCAT, you may also encounter R given as 8.314 J/(K • mol), which is derived when SI units of pascal (for pressure) and cubic meters (for volume) are substituted into the ideal gas law. Although the value(s) for R will be given to you on Test Day (as will the values for almost all constants), it is important that you learn to recognize the appropriate value for R based on the units of the variables as they are given to you in the passage or question stem. The variables in the law itself become easy to remember if you “sound out” the law: “piv-nert”

**Example:** What volume would 12 g of helium occupy at 20°C and a pressure of 380 mm Hg?

**Solution:** The ideal gas law can be used, but first, all of the variables must be converted to yield units that will correspond to the expression of the gas constant as 0.0821 L • atm/(mol • K).

The ideal gas law is useful to you not only for standard calculations of pressure, volume, or temperature of a gas under a set of given conditions but also for determinations of gas density and molar mass.

**Density**

We define density as the ratio of the mass per unit volume of a substance and, for gases, express it in units of grams per liter (g/L). The ideal gas law contains variables for volume and number of moles, so we can rearrange the law to calculate the density of any gas:

*PV* = *n*R*T*

where

Therefore,

and

Because success on the MCAT depends on your ability to think critically, analyze the information provided to you, and discern which of it is necessary and useful and what’s merely a “red herring,” you should work to become comfortable in approaching problems from different angles, thereby ensuring that you will have many “tools” in your Test Day “tool belt.” As an example, let’s consider a second approach to determining the density of a gas that could prove useful to you on the MCAT.

For this approach, we need to start with the volume of a mole of gas at STP, which is 22.4 L. We will then calculate the effect of pressure and temperature on the volume (to the degree that they differ from STP conditions). Finally, we’ll calculate the density by dividing the mass by the new volume. The following equation can be used to relate changes in temperature, volume, and pressure of a gas:

where the subscripts 1 and 2 refer to the two states of the gas (at STP and at the conditions of actual temperature and pressure). If you look carefully at this equation, you’ll notice that this assumes that the number of moles of gas is held constant, and in fact, we could write the equation as follows:

To calculate a change in volume, the equation is rearranged as follows:

*V*_{2} is then used to find the density of the gas under nonstandard conditions:

On Test Day, you may find it helpful to visualize how the changes in pressure and temperature affect the volume of the gas, and this can serve as a check to make sure that you have not accidentally switched the values of pressure and temperature in the numerator and denominator of the respective pressure and temperature ratios. For example, you would be able to predict (without even doing the math) that doubling the temperature would result in doubling the volume, and doubling the pressure would result in halving the volume, so doubling both at the same time results in a final volume that is equal to the original volume.

**Example:** What is the density of HCl gas at 2 atm and 45°C?

**Solution:** At STP, a mole of gas occupies 22.4 liters. Because the increase in pressure to 2 atm decreases volume, 22.4 L must be multiplied by . Because the increase in temperature increases volume, the temperature factor will be .

**Molar Mass**

Sometimes the identity of a gas is unknown, and the molar mass (__Chapter 4__, Compounds and Stoichiometry) must be determined in order to identify it. Using the equation for density derived from the ideal gas law, we can calculate the molar mass of a gas in the following way. The pressure and temperature of a gas contained in a bulb of a given volume are measured, and the mass of the bulb plus sample is measured. Then, the bulb is evacuated (the gas is removed), and the mass of the empty bulb is determined. The mass of the bulb plus sample minus the mass of the evacuated bulb yields the mass of the sample. Finally, the density of the sample is determined by dividing the mass of the sample by the volume of the bulb. This gives the density at the particular conditions of the given temperature and pressure. Using *V*_{2} = *V*_{1} (*P*_{1}/*P*_{2})(*T*_{2}/*T*_{1}), we then calculate the volume of the gas at STP (substituting 273 K for *T*_{2} and 1 atm for *P*_{2}). The ratio of the sample mass divided by *V*_{2} gives the density of the gas at STP. The molar mass can then be calculated as the product of the gas’s density at STP and the STP volume of any gas, 22.4 L/mol.

**Example:** What is the molar mass of a 2 L sample of gas that weighs 8 g at a temperature of 15°C and a pressure of 1.5 atm?

**SPECIAL CASES OF THE IDEAL GAS LAW**

Now that we have considered the ideal gas law as the mathematical relationship between four variables that define the state of a gas (pressure, volume, temperature, and moles of gas), we can examine two special cases of the ideal gas law in which some of the variables are held constant as the gas system undergoes a process. Even though the following two laws were developed before the ideal gas law, it is conceptually helpful to understand them as simple special cases of the more general ideal gas law.

**Boyle’s Law**

Robert Boyle conducted a series of experimental studies in 1660 that led to his formulation of a law that now bears his name: Boyle’s law. His work showed that for a given gaseous sample held at constant temperature (isothermal conditions) the volume of the gas is inversely proportional to its pressure:

*PV* = k, or *P*_{1}*V*_{1} = *P*_{2}*V*_{2}

where k is a proportionality constant and the subscripts 1 and 2 represent two different sets of pressure and volume conditions. Careful examination of Boyle’s law shows that it is, indeed, simply the special case of the ideal gas law in which *n,* R, and *T* are constant:

*PV* = *n*R*T* = constant

A plot of volume versus pressure for a gas is shown in Figure 7.2.

**Key Concept**

Boyle’s law is a derivation of the ideal gas law and states that pressure and volume are inversely related: When one increases, the other decreases.

**Figure 7.2**

**MCAT Expertise**

Sometimes it is easier to remember the shape of the graph to help you recall the variables’ relationship on Test Day. Here we can see that as pressure increases, the volume decreases, and vice versa. These ratios and relationships will often answer questions on the MCAT without your having to do any math.

**Example:** Under isothermal conditions, what would be the volume of a 1 L sample of helium if its pressure is changed from 12 atm to 4 atm?

**Solution:**

*P*_{1} = 12 atm *P*_{2} = 4 atm

*V*_{1} = 1 L *V*_{2} = *X*

*P*_{1}*V*_{1} = *P*_{2}*V*_{2}

12 atm (1 L) = 4 atm ( *X* )

L = *X**X* = 3 L

**Law of Charles and Gay-Lussac**

In the early 19th century Gay-Lussac published findings based, in part, on earlier unpublished work by Charles; hence, the law of Charles and Gay-Lussac is more commonly known simply as Charles’ law. The law states that at constant pressure, the volume of a gas is proportional to its absolute temperature, expressed in degrees Kelvin. (Remember the conversion from Celsius to Kelvin is *T*_{K} = *T*_{°C} + 273.15.) Expressed mathematically, Charles’ law is

where, again, k is a proportionality constant and the subscripts 1 and 2 represent two different sets of temperature and volume conditions. Careful examination of Charles’ law shows that it is another special case of the ideal gas law in which *n,* R, and *P* are constant:

**Key Concept**

Charles’ law is also a derivation of the ideal gas law and states that volume and temperature are directly proportional: When one increases, the other does too.

A plot of temperature versus volume is shown in __Figure 7.3__. It is interesting to note that if one extrapolates the *V* versus *T* plot for a gas back to where *V* = 0 (as it should for an ideal gas), we find that *T* 0 K!

**Real World**

While the temperature of 0 K cannot be physically attained, curves such as this one were originally used to figure out its location.

__Figure 7.3__

**Example:** If the absolute temperature of 2 L of gas at constant pressure is changed from 283.15 K to 566.30 K, what would be the final volume?

**Solution:**

**DALTON’S LAW OF PARTIAL PRESSURES**

When two or more gases are found in one vessel without chemical interaction, each gas will behave independently of the other(s). That is to say that each gas will behave as if it were the only gas in the container. Therefore, the pressure exerted by each gas in the mixture will be equal to the pressure that gas would exert if it were the only one in the container. The pressure exerted by each individual gas is called the partial pressure of that gas. In 1801, John Dalton derived an expression, now known as Dalton’s law of partial pressures, which states that the total pressure of a gaseous mixture is equal to the sum of the partial pressure of the individual components. The equation for Dalton’s law is

*P*_{T} = *P*_{A} + *P*_{B} + *P*_{C} + · · ·

The partial pressure of a gas is related to its mole fraction and can be determined using the following equations:

*P*_{A} = *P*_{T}*X*_{A}

where

*X* _{A} = (moles of A / total moles of all gases)

**Key Concept**

When more than one gas is in a container, each contributes to the whole as if it were the only gas present. So add up all of the pressures of the individual gases, and you get the whole pressure of the system.

**Example:** A vessel contains 0.75 mol of nitrogen, 0.20 mol of hydrogen, and 0.05 mol of fluorine at a total pressure of 2.5 atm. What is the partial pressure of each gas?

First calculate the mole fraction of each gas.

Then calculate the partial pressure.

*P*_{A} = *X*_{A}*P*_{T}

**Real Gases**

Throughout our discussion of the laws and theory that describe and explain the behavior of gases, we have stressed that the fundamental assumption is a gas that behaves ideally. However, our world is not one of ideal gases but rather real ones, and real gases have volumes and interact with each other in measurable ways. In general, the ideal gas law is a good approximation of the behavior of real gases, but all real gases deviate from ideal gas behavior to some extent, particularly when the gas atoms or molecules are forced into close proximity under high pressure and at low temperature. Under these “nonideal” conditions, the molecular volume and intermolecular forces become significant.

You can think of these nonideal conditions as the degree to which human populations are “forced” to interact with each other in high-population density regions. In 2007, for example, the population density of Washington, D.C., was 9,581 people per square mile, whereas that of Alaska (the least densely populated state in the Union at that time) was 1.2 people per square mile. Quite literally, the personal space of an individual living in the nation’s capital cannot be ignored, whereas the vast physical separation between people living in Alaska (on average) makes the notion of “personal space” so insignificant as to be almost laughable. Continuing the human-as-real-gas analogy, how often do people say to each other on a hot and humid day, “Ugghh, don’t come near me. It’s too hot!”? And how many romantic songs include imagery of lovers snuggling together on a cold evening trying to keep warm?

**MCAT Expertise**

At high temperature and low pressure, deviations from ideality are usually small; good approximations can still be made from the ideal gas law.

**DEVIATIONS DUE TO PRESSURE**

As the pressure of a gas increases, the particles are pushed closer and closer together. As the condensation pressure for a given temperature is approached, intermolecular attraction forces become more and more significant, until the gas condenses into the liquid state (see “Gas-Liquid Equilibrium” in __Chapter 8__).

**MCAT Expertise**

On the MCAT, an understanding of nonideal conditions will help with determining how gases’ behavior may deviate.

At moderately high pressure (a few hundred atmospheres), a gas’s volume is less than would be predicted by the ideal gas law due to intermolecular attraction. At extremely high pressure, however, the size of the particles becomes relatively large compared to the distance between them, and this causes the gas to take up a larger volume than would be predicted by the ideal gas law.

**DEVIATIONS DUE TO TEMPERATURE**

As the temperature of a gas is decreased, the average velocity of the gas molecules decreases, and the attractive intermolecular forces become increasingly significant. As the condensation temperature is approached for a given pressure, intermolecular attractions eventually cause the gas to condense to a liquid state (see “Gas-Liquid Equilibrium” in __Chapter 8__).

As the temperature of a gas is reduced toward its condensation point (which is the same as its boiling point), intermolecular attraction causes the gas to have a smaller volume than that which would be predicted by the ideal gas law. The closer the temperature of a gas is to its boiling point, the less ideal is its behavior.

__VAN DER WAALS EQUATION__** OF STATE**

**Key Concept**

Note that if a and b are both zero, the van der Waals equation reduces to the ideal gas law.

There are several gas equations, or gas laws, that attempt to correct for the deviations from ideality that occur when a gas does not closely follow the ideal gas law. The van der Waals equation of state is one such equation:

where a and b are physical constants experimentally determined for each gas. The a term corrects for the attractive forces between molecules (*a* for *attractive*) and as such will be smaller for gases that are small and less polarizable (such as helium), larger for gases that are larger and more polarizable (such as Xe or N_{2}), and largest for polar molecules such as HCl and NH_{3}. The b term corrects for the volume of the molecules themselves. Larger values of b are thus found for larger molecules. Numerical values for a are generally much larger than those for b.

**Example:** Find the correction in pressure necessary for the deviation from ideality for 1 mole of ammonia in a 1 liter flask at 0°C. (For NH_{3}, a = 4.2, b = 0.037)

**Solution:** According to the ideal gas law,

*P* = *n*R*T*/*V* = (1)(0.0821)(273)/(1) = 22.4 atm, while according to the van der Waals equation,

The pressure is thus 3.3 atm less than would be predicted from the ideal gas law, or an error of 15 percent.

Note that including the correction term (a) has the effect of increasing the observed pressure (*P*) to that predicted by the ideal gas law.

**MCAT Expertise**

Be familiar with the concepts embodied by this equation but do not spend too much time working with it or memorizing it, as it is not likely to be tested directly on the MCAT.

**Conclusion**

In this chapter, we reviewed the basic characteristics and behaviors of gases. The kinetic molecular theory of gases lays out the explanation for the behavior of ideal gases as described by the ideal gas law. The ideal gas law shows the mathematical relationship among four variables associated with gases: pressure, volume, temperature, and number of moles. We examined special cases of the ideal gas law in which temperature (Boyle’s law) or pressure (Charles’ law) is held constant. Boyle’s law shows that when temperature is held constant, there is an inverse relationship between pressure and volume. Charles’ law shows that when pressure is held constant, there is a direct relationship between temperature and volume. We also examined Dalton’s law, which relates the partial pressure of a gas to its mole fraction and the sum of the partial pressures of all the gases in a system to the total pressure of the system. Finally, we examined the ways in which real gases deviate from the predicted behaviors of ideal gases. The van der Waals equation of state is a useful equation for correcting deviations based on molecular interactions and volumes.

From helium-filled balloons to the bubbles of carbon dioxide in a glass of soda, from the pressurized gases used for scuba diving to the air we breathe on land, gases are all around us. And yet, for all the different gases that bubble, flow, and settle in and through our daily living experiences, they behave in remarkably similar ways. Expect that the MCAT will treat gases with the level of attention that is appropriate to their importance in our physical lives.

**CONCEPTS TO REMEMBER**

Gases are the least dense phase of matter. They are classified, along with liquids, as fluids because they flow in response to shearing forces and conform to the shape of their containers. Unlike liquids, however, gases are compressible.

The state of a gas system can be characterized by four properties: pressure, volume, temperature, and number of moles. Standard temperature and pressure (STP) is a set of conditions common in the study of gases; standard temperature is 273 K (0°C), and standard pressure is 1 atm.

Ideal gases are described by the kinetic molecular theory of gases, which characterizes gases as composed of particles with negligible volume, with no intermolecular forces, in continuous and random motion, undergoing elastic collisions with each other and the walls of their container, and having an average kinetic energy that is proportional to the temperature.

Graham’s law of diffusion and effusion states that for two or more gases at the same temperature, a gas with lower molar mass will diffuse or effuse more rapidly than a gas with higher molar mass.

Regardless of chemical identity, equal amounts of gases occupy the same volume if they are at the same temperature and pressure. For example, one mole of any gas occupies 22.4 liters at STP.

The ideal gas law, *PV* = *n*R*T,* describes the mathematical relationship among the four variables of the gas state for an ideal gas.

Boyle’s law is a special case of the ideal gas law for which temperature is held constant; it shows an inverse relationship between pressure and volume.

Charles’ law is a special case of the ideal gas law for which pressure is held constant; it shows a direct relationship between temperature and volume.

Dalton’s law of partial pressure states that the individual gas components of a mixture of gases will exert individual pressures, called partial pressures, in proportion to their mole fractions. The total pressure of a mixture of gases is equal to the sum of the individual partial pressures of the individual gas components.

The behavior of real gases deviates from that predicted by the ideal gas law, especially under conditions of very high pressure or very low temperature. The van der Waals equation of state is used to correct for deviations due to intermolecular attractions and molecular volumes.

**EQUATIONS TO REMEMBER**

__Practice Questions__

**1.** Based on your knowledge of gases, what conditions would be least likely to result in ideal gas behavior?

A. High pressure and low temperature

B. Low temperature and large volume

C. High pressure and large volume

D. Low pressure and high temperature

**2.** Calculate the density of neon gas at STP in g L^{-1}. The molar mass of neon can be approximated to 20.18 g mol^{-1}.

A. 452.3 g L^{-1}

B. 226.0 g L^{-1}

C. 1.802 g L^{-1}

D. 0.9009 g L^{-1}

**3.** A leak of helium gas through a small hole occurs at a rate of 3.22 × 10^{-5} mol s^{-1}. Compare the leakage rates of neon and oxygen gases to helium at the same temperature and pressure.

A. Neon will leak faster than helium; oxygen will leak slower than helium.

B. Neon will leak faster than helium; oxygen will leak slower than helium.

C. Neon will leak slower than helium; oxygen will leak slower than helium.

D. Neon will leak slower than helium; oxygen will leak faster than helium.

**4.** A 0.040 gram piece of magnesium is placed in a beaker of hydrochloric acid. Hydrogen gas is generated according to the following equation:

Mg (*s*) + 2 HCl (*aq*) MgCl_{2} (*aq*) + H_{2} (*g*)

The gas is collected over water at 25°C, and the pressure during the experiment reads 784 mm Hg. The gas displaces a volume of 100 mL. The vapor pressure of water at 25°C is approximately 24.0 mm Hg. From this data, calculate how many moles of hydrogen are produced.

A. 4.22 × 10^{-3} moles hydrogen

B. 4.09 × 10^{-3} moles hydrogen

C. 3.11 moles hydrogen

D. 3.20 moles hydrogen

**5.** The properties of ideal gases state that ideal gases

I. have no volume.

II. have no attractive forces between them.

III. have no mass.

A. I, II, and III

B. I only

C. I and II only

D. I and III only

**6.** An 8.01 g sample of NH_{4}NO_{3} (*s*) is placed into an evacuated 10.00 L flask and heated to 227°C. After the NH_{4}NO_{3} totally decomposes, what is the approximate pressure in the flask?

NH_{4}NO_{3} (*s*) N_{2}O (*g*) + H_{2}O (*g*)

A. 0.600 atm

B. 0.410 atm

C. 1.23 atm

D. 0.672 atm

**7.** The kinetic molecular theory states that

A. the average kinetic energy of a molecule of gas is directly proportional to the temperature of the gas in Kelvin.

B. collisions between gas molecules are inelastic.

C. elastic collisions result in a loss of energy.

D. all gas molecules have the same kinetic energy.

**8.** The plots of two gases at STP are shown below. One of the gases is 1.0 L of helium, and the other is 1.0 L of bromine. Which plot corresponds to each gas and why?

A. Curve A is helium and curve B is bromine, because helium has a smaller molar mass than bromine.

B. Curve A is helium and curve B is bromine, because the average kinetic energy of bromine is greater than the average kinetic energy of helium.

C. Curve A is bromine and curve B is helium, because helium has a smaller molar mass than bromine.

D. Curve A is bromine and curve B is helium, because the average kinetic energy of bromine is greater than the average kinetic energy of helium.

**9.** A balloon at standard temperature and pressure contains 0.20 moles of oxygen and 0.60 moles of nitrogen. What is the partial pressure of oxygen in the balloon?

A. 0.20 atm

B. 0.30 atm

C. 0.60 atm

D. 0.25 atm

**10.** The temperature at the center of the sun can be estimated based on the approximation that the gases at the center of the sun have an average molar mass equal to 2.00 g/mole. Approximate the temperature at the center of the sun using these additional values: The pressure equals 1.30 × 10^{9}atm, and the density at the center equals 1.20 g/cm^{3}.

A. 2.6 × 10^{7} K

B. 2.6 × 10^{10} K

C. 2.6 × 10^{4} K

D. 2.6 × 10^{6} K

**11.** The gaseous state of matter is characterized by the following properties:

I. Gases are compressible.

II. Gases assume the volume of their container.

III. Gas particles exist as diatomic molecules.

A. I and II only

B. I and III only

C. III only

D. I, II, and III

**12.** A gas at a temperature of 27°C has a volume of 60.0 mL. What temperature change is needed to increase this gas to a volume of 90.0 mL?

A. A reduction of 150°C

B. An increase of 150°C

C. A reduction of 40.5°C

D. An increase of 40.5°C

**13.** A gaseous mixture contains nitrogen and helium and has a total pressure of 150 torr. The nitrogen particles comprise 80 percent of the gas, and the helium particles make up the other 20 percent of the gas. What is the pressure exerted by each individual gas?

A. 100 torr nitrogen, 50.0 torr helium

B. 120 torr nitrogen, 30.0 torr helium

C. 30.0 torr nitrogen, 150 torr helium

D. 50.0 torr nitrogen, 100 torr helium

**14.** In which of the following situations is it impossible to predict how the pressure will change for a gas sample?

A. The gas is cooled at a constant volume.

B. The gas is heated at a constant volume.

C. The gas is heated, and the volume is simultaneously increased.

D. The gas is cooled, and the volume is simultaneously increased.

**Small Group Questions**

**1.** How does the addition or removal of gas B from a vessel affect the partial pressure of gas A?

**2.** Is hydrogen bonding present in steam?

**Explanations to Practice Questions**

**1. A**

Gases deviate from ideal behavior at higher pressures, which forces molecules closer together. The closer they are, the more they can participate in intermolecular forces, which violate the definition of an ideal gas. As the temperature of a gas is reduced, the average velocity of the gas molecules decreases, and the attractive intermolecular forces become more significant. This results in the loss of another characteristic of an ideal gas, and thus less ideal behavior. Answer choices that include the opposite, high temperature and low pressure, are incorrect because gases behave most ideally under these conditions. At high temperatures, molecules will move quickly and exhibit random motion and elastic collisions, which is a property of ideal gases. In an ideal gas it is assumed that there are no intermolecular attractions between gas molecules, which is valid at low pressures when there is ample space between them.

**2. D**

Density equals mass divided by volume. The mass of 1 mole of neon gas equals 20.18 grams. At STP, 1 mole of neon occupies 22.4 L. Dividing the mass, 20.18 grams, by the volume, 22.4 L, gives an approximate density of 0.9009 g L^{-1}.

**3. C**

Graham’s law of effusion states that the relative rates of effusion of two gases at the same temperature and pressure are given by the inverse ratio of the square roots of the masses of the gas particles. In equation form, Graham’s law can be represented by: *Rate*_{1} /*Rate*_{2} = . If a molecule has a higher molecular weight, then it will leak at a slower rate than a gas with a lower molecular weight. Both neon and oxygen gases will leak at slower rates than helium because they both weigh more than helium.

**4. B**

The pressure of the gas is calculated by subtracting the vapor pressure of water from the measured pressure during the experiment: 784 mm Hg - 24 mm Hg = 760 mm Hg, or 1 atm. The ideal gas law can be used to calculate the moles of hydrogen gas. The volume of the gas equals 0.100 L, the temperatures equals 298 K, and R = 0.0821 (L atm / mol K). Solving the equation *PV* = *n*R*T* for *n* gives 4.09 × 10^{-3} moles of hydrogen. (A) is incorrect and would result from mistakenly using 784 mm Hg of the pressure instead of the pressure adjusted for water vapor. (C) and (D) would both result from incorrectly using a pressure in mm Hg instead of converting to atm while using the gas constant R = 0.0821 (L atm / mol K).

**5. C**

Ideal gases are said to have no attractive forces between molecules. They are considered to have point masses, which theoretically take up no volume.

**6. C**

The first thing to do is balance the given chemical equation. The coefficients, from left to right, are 1, 1, and 2. The mass of solid, 8.01 grams, can be converted to moles of gas product by dividing by the molar mass of NH_{4}NO_{3}(*s*) (80.06 g) and multiplying by the molar ratio of 3 moles of gas product to one mole of NH_{4}NO_{3}(*s*). This gives approximately 0.300 moles of gas product. The ideal gas equation can be used to obtain the pressure in the flask. The values are as follows: R equals 0.0821 (L atm / mole K), the temperature in Kelvin is 500 K, and the volume is 10.00 L. Solving for*P* in the equation *PV* = *n*R*T* gives a pressure of about 1.23 atm.

**7. A**

The average kinetic energy is directly proportional to the temperature of a gas in Kelvin. The kinetic molecular theory states that collisions between molecules are elastic and thus do not result in a loss of energy. The kinetic energy of each gas molecule is not the same.

**8. C**

At STP, the difference between the distribution of velocities for helium and bromine gas is due to the difference in molar mass (*Rate* /*Rate* ) = . Helium has a smaller molar mass than bromine. Particles with small masses travel faster than those with large masses, so the helium gas corresponds to curve B with higher velocities. (A) and (B) are incorrect because they inaccurately identify each curve on the graph. Given that the gases are at the same temperature (STP), we can recall that temperature relates to kinetic energy (KE). *KE* = k*T* . The gases average KE, therefore should be the same. Therefore, answer (D) is also incorrect.

**9. D**

At STP, the pressure inside the balloon equals 1 atm. The total number of moles in the balloon equals 0.20 moles plus 0.60 moles, or 0.80 moles. *P*_{O2} equals the mole fraction of oxygen (0.20/0.80) times the total pressure, 1 atm. The partial pressure of oxygen is 0.25 atm.

**10. A**

The ideal gas law can be modified to include density and determine the temperature of the sun.

*n* = mass/molecular weight *density* (denoted *D* for this problem) = *mass*/*V*

The density is given in g/cm^{3} and must be converted to g/L so the units cancel in the above equation. Because 1 cm^{3} equals 1 mL and there are 1,000 mL in 1 L, the density can be multiplied by 1,000 to be converted to g/L.

**11. A**

Gases are compressible, because they travel freely with large amounts of space between molecules. Because gas particles are far apart from each other and in rapid motion, they tend to take up the volume of their container. Many gases that exist as diatomic molecules (i.e., O_{2}, H_{2}, N_{2}), but this is not a property that characterizes all gases.

**12. B**

We will use *V*_{1}/*T*_{1} = *V*_{2}/*T*_{2}. First, we must convert the temperature to Kelvin by adding 273 to get 300 K as the initial temperature. Plugging into the equation and solving for *T*_{2} gives 450 K. Subtracting the initial temperature, 300 K, gives an increase of 150 K. They are measured by the same increments, so an increase of 150 K corresponds to an increase of 150°C.

**13. B**

The partial pressure of each gas is found by multiplying the total pressure by the mole fraction of the gas. Because 80 percent of the molecules are nitrogen, the mole fraction of nitrogen gas is equal to 0.80. Similarly, for helium, the mole fraction is 0.20, because helium comprises 20 percent of the gas molecules. To find the pressure exerted by nitrogen, multiply the total pressure (150 torr) by 0.80 to obtain 120 torr of nitrogen. To find the pressure exerted by helium, multiply the total pressure by 0.20 to get 30 torr of helium.

**14. C**

### Both a change in temperature and a change in volume can affect the gas’s pressure. So if one of those two variables is kept constant (i.e., (A) and (B)), we’ll definitely be able to predict which way the pressure will change. At a constant volume, heating the gas will increase its pressure, and cooling the gas will decrease it. What about when both temperature and volume are changing? If they have the same effect on pressure, then we can still predict which way it will change. This is the case in (D). Cooling the gas and increasing its volume both decrease pressure. (C), on the other hand, presents us with too vague a scenario for us to predict definitively the change in pressure. Heating the gas would amplify the pressure, while increasing the volume would decrease it. Without knowing the magnitude of each influence, it’s impossible to say whether the pressure would increase, decrease, or stay the same.