Kaplan MCAT General Chemistry Review

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 P₁V₁ = P₂V₂

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 = nRT = 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:

P₁ = 12 atm P₂ = 4 atm

V₁ = 1 L V₂ = X

P₁V₁ = P₂V₂

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In the early 19th century Gay-Lussac published findings based, in part, on earlier unpublished work by Charles; hence, the law of Charles and Gay-Lussac is more commonly known simply as Charles’ law.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_TX_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.