The Ideal Gas Law

The ideal gas law is a simple model that allows us to predict the behavior of gases in the world. It is a combination of the previous laws that we have studied (Boyle's, Charles', Avogadro's). Rather than simply looking at proportionalities, it relates pressure, volume, absolute temperature, and the number of moles quantitatively with a universal constant $R$ that we call the ideal gas constant.

\[P\,V=nR\,T\]

The value for R will depend on what units you are using for the properties of the gas. The ideal gas law has many implications that will be discussed below. However, the most remarkable aspect is that the same model works quite well for all gases (N₂, He, C₃H₈, ...).

video here

Universal Gas Constant

The ideal gas constant is a Universal constant that we use to quantify the relationship between the properties of a gas. The constant $R$ that we typically use relates pressure in atmospheres, volume in liters, and temperature in Kelvin. In this case, it has the value and units of

\[R=0.08206 {\rm \;\;L\;atm\;mol^{-1}\;K^{-1}}\]

The gas constant $R$ will appear in many contexts as this is a Universal constant that relates energy and temperature. A pressure times a volume is an energy. As such, you will also encounter the gas constant $R$ in typical energy units of Joules

\[R=8.314 {\rm \;\;J\;mol^{-1}\;K^{-1}}\]

Finally, ignore the blunder in the video where Dr. Vanden Bout can't remember the correct value for $R$ with units of atm. The correct value is 0.08206 atm L mol^-1 K^-1

Standard Conditions

Often in the laboratory we would like to have conditions (state functions) that are similar to those used by scientists in other laboratories. As such, as a community, we have decided upon some conditions that we will call "standard". The two state functions we can typically control are temperature ( $T$ ) and pressure ( $P$ ), so these are the conditions we standardize. We will call these conditions "Standard Temperature and Pressure" or STP. We will take standard pressure as 1 atm, and standard temperature as 0°C. You should note that these are choices. Like many things in life it is difficult to get everyone to agree. Recently, the International Union of Pure and Applied Chemistry (IUPAC) adopted 1 bar as standard pressure and 25°C as standard temperature. This "new" standard is referred to as Standard Ambient Temperature and Pressure (SATP). However, as this new condition has been slow to catch on, we will stick with the "old" standard (STP) unless told otherwise.

Keep them Straight

STP : 0°C (273.15 K) and 1 atm ←(our default "standard" in textbook and Quest)

SATP : 25°C (298.15 K) and 1 bar

Number Density

Number density is a useful concept for thinking about macroscopic samples in a microscopic way. Chemists often try to "visualize" materials with a molecular perspective. Number density can be thought of as the number of particles that are present in a particular volume. As these numbers can be very very large, we typically think of this as the number of moles (a fixed number of particles) within a given volume. This quantity is

$${n\over V}={P\over {RT}}$$

So, if we have any two gas samples that are behaving ideally, they have the same number of particles per volume when the temperature and pressure are the same. For example, if I had two balloons in a room, they would have the same pressure (approximately one atmosphere) and the same temperature (whatever the temperature was in the room). Therefore, they would have the same number density. If the balloons had the same volume then they would have identical numbers of particles. This is really a different way of stating Avogadro's Law.

Molar Volume

Molar volume is another way to view number density. The molar volume is the volume of 1 mole of substance. It can be simply found by dividing the volume of a sample by the number of moles of that sample

\[V_m = {V\over n}\]

This is simply the inverse of the number density. It is a useful quantity to "think about" things from a molecular perspective. As all gases that are behaving ideally have the same number density, they will all have the same molar volume. At STP this will be 22.42 L. This is useful if you want to envision the distance between molecules in different samples. For instance if you have a sample of liquid water, it has a mass density of 1 g mL^-1. Since water has a molecular weigh of 18 g mol^-1, 1 mole will have a mass of 18 g. From the density this should have a volume of only 18 mL. Clearly this is a lot less than an ideal gas at STP which is more than 1000 times larger in terms of its molar volume.

The video below is a further discuss of Standard Molar Volume (molar volume at STP).