# Density Calculations

## A derivation of the ideal gas equation allows us to use density calculations to evaluate the behaviors of ideal gases of unknown quantity.

#### Key Points

• Using density calculations allows us to evaluate the behaviors of gases of unknown volume.

• We can determine the density of an ideal gas using knowledge of three properties of the ideal gas being evaluated.

• This form of the ideal gas equation relates pressure, density, and temperature of an ideal gas independent of the quantity of gas.

#### Terms

• a measure of the amount of matter contained by a given volume

#### Figures

1. ##### Atmosphere Composition

Atmospheric science offers us one plausible real life application of the density form of the ideal gas equation. Earth's atmosphere is composed of gases that support life on our planet. The density form of the ideal gas law enables us to study the behavior of these gases without requiring that the gases be enclosed in a container of known volume.

2. ##### Astronomical Applications of the Ideal Gas Law

The Taurus Molecular Cloud consists of dust and various gases including hydrogen and helium. The density form of the ideal gas equation may be of theoretical use when studying such astronomical phenomena as star formation.

3. ##### Ideal Gas Law Practice Problems with Density

Instead of using the regular ideal gas equation, PV=nRT, we'll use a transformed version (D=PM/RT) in order to solve a problem with density and molar mass.

The ideal gas equation in the form PV=nRT is an excellent tool for understanding the relationship between the pressure, volume, amount, and temperature of an ideal gas in a defined environment where we are able to control for constant volume. However, the ideal gas equation, in its most common form, is not useful when we seek to to examine the behavior of gases of undetermined volume, such as the gases in the clouds that surround the stars in our solar system or the atmospheric gases that support life on our planet Figure 1Figure 2. For the purposes of studying gases with volumes that are difficult to quantify, we can make use of calculations employing the physical property of density, defined as the mass of the gas per unit volume, to derive a form of the ideal gas equation that has broader applications.

n = # moles of gas = mass of gas (m)/molecular weight (M)

We know the ideal gas equation in the form $PV=nRT$. We also know that:

and that:

$density = \frac{m}{V}$.

If we substitute m/M for n:

$PV= \frac{m}{M}RT$

and simplify, we arrive at:

$P/RT=\frac{m}{M}/{V}$

If we simplify further, we get:

$d=\frac{MP}{RT}$

This derivation of the ideal gas equation allows us to characterize the relationship between the pressure, density, and temperature of the gas sample independent of the volume the gas occupies. It allows us to determine the density of a gas sample given its pressure and temperature or to determine the molar mass of a gas sample given its density.

Figure 3

#### Key Term Glossary

constant
Consistently recurring over time; persistent
##### Appears in these related concepts:
density
a measure of the amount of matter contained by a given volume
##### Appears in these related concepts:
gas
Matter in a state intermediate between liquid and plasma that can be contained only if it is fully surrounded by a solid (or held together by gravitational pull); it can condense into a liquid, or can (rarely) become a solid directly.
##### Appears in these related concepts:
ideal gas
a hypothetical gas whose molecules exhibit no interaction and undergo elastic collision with each other and with the walls of the container
##### Appears in these related concepts:
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles.
##### Appears in these related concepts:
ideal gas equation
The ideal gas equation is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions.
##### Appears in these related concepts:
Ideal Gas Equation
The Ideal Gas Law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations.  The most commonly written form is PV=nRT.
##### Appears in these related concepts:
mass
The quantity of matter that a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.
##### Appears in these related concepts:
molar mass
the mass of a given substance (chemical element or chemical compound) divided by its amount of substance
##### Appears in these related concepts:
mole
In the International System of Units, the base unit of the amount of substance; the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. Symbol: mol. The number of atoms in a mole is known as Avogadro’s number.
##### Appears in these related concepts:
physical property
A physical property is any property that is measurable whose value describes a physical system's state.
##### Appears in these related concepts:
Pressure
the amount of force that is applied over a given area divided by the size of this area
##### Appears in these related concepts:
system
the part of the universe being studied, arbitrarily defined to any size desired
##### Appears in these related concepts:
temperature
A measure of cold or heat, often measurable with a thermometer.
##### Appears in these related concepts:
volume
A unit of three-dimensional measure of space that comprises a length, a width, and a height. It is measured in units of cubic centimeters in metric, or cubic inches or cubic feet in English measurement.