Kaplan MCAT General Chemistry Review

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 = nRT

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:

V₂ 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 V₂ = V₁ (P₁/P₂)(T₂/T₁), we then calculate the volume of the gas at STP (substituting 273 K for T₂ and 1 atm for P₂). The ratio of the sample mass divided by V₂ 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?