HS Chemistry - Intermolecular Forces

The Ideal Gas Law

Overview of The Page

This page will cover:

What is the ideal gas law?
How can the ideal gas law be derived?
What are some exceptions to the ideal gas law?

All gases possess some properties. Some gases are more true to these properties than others. Some gases represent some properties better than some properties. Is there a gas that represents all of them perfectly?

The ideal gas is the gas which remains absolutely true to all of these properties. These properties are:

A gas is a group of small particles moving randomly.

There are no intermolecular forces of attraction between the particles.

The particles are so small, and the distance between them so great, that the size of the particles can be ignored - we can assume it is 0.

There is no energy lost when the particles collide with each other or the walls of the container.

All particles have the exact same kinetic energy (all particles have the exact same thermal energy)

In reality, the gases we observe, called real gases, don't represent all of these properties perfectly. For example, there will always be some amount of intermolecular force between the particles, and the particles do take up a small amount of space. And not all particles have the exact same kinetic energy.

But all gases will have properties that are really close to those of the ideal gas. For example, while there is some intermolecular force between the particles, it is really small. And we can use the average of the particle's kinetic energy $the temperature of the gas$ and use that instead.

We can use the ideal gas, and its properties, as a baseline for dealing with real gases, since real gases are really close to ideal gases, and all gases trend towards the ideal gas.

The ideal gas law can be used to find the properties of an ideal gas. Since real gases are really close to ideal gases, the ideal gas law can also be used to find the properties of a real gas. In short, the ideal gas law can be used to find the properties of a gas.

The ideal gas law is:

PV = nRT

Where:

P is pressure $in atm$
V is volume $in Liters$
n is the number of particles $in moles$
R is the Universal Gas Constant, which remains the same for all gases
T is the temperature $in Kelvin$

Given any three of P, V, n, or T, we can calculate the fourth (since R will always be the same, we already know it). However, before we perform calculations, we must change them to the correct units.

Variable	Unit Conversion
P $Pressure$	1 atm (atmospheric pressure) = 101.3 kPa (kiloPascals) 1 atm = 760 mm Hg (760 mm is the distance that mercury (Hg) will be pushed down by a gas with a pressure of 1 atm in a barometer)
V $Volume$	1.0 L = 1.0 dm³
n $number of moles$	We don't need to worry about converting the number of moles, as it will always be given as a whole number
R $Universal Gas Constant$	R has the same value for all gases If P (Pressure) is given in atm, use R = 0.0821 (atm L)/(mol K) If P (Pressure) is given in kPa, use R = 8.314 (kPa L)/(mol K)
T $Temperature$	To convert Celsius to Kelvin, add 273.15 This is because 0°C = 273.15K, and an increase of 1°C is equal to an increase of 1K To convert Fahrenheit to Kelvin, first convert Fahrenheit to Celsius, then convert Celsius to Kelvin To convert Fahrenheit (F) to Celsius (C), C = 5 × (F − 32) ÷ 9

Using the Ideal Gas Law, we can derive two things:

1 mole of any gas at room temperature takes up 24.0 dm³ of volume
1 mole of any gas at 0°C takes up 22.4 dm³ of volume

Deriving the Ideal Gas Law

So you're probably wondering how we got the equation for the Ideal Gas Law. It comes from combining four different laws about gases.

Boyle's Law states that if we have a fixed temperature, then pressure and volume will be inversely proportional to one another.
1. If the temperature remains constant the same, then the particles move at the same speed.
2. If we increase the volume of the container, then the particles will have more space to move. They'll have to traverse more space to reach the containers of the wall, and it'll take them longer to do so. Fewer particles will collide against the wall in a given period of time. The particles will exert less force over a given area in a given amount of time. Thus, the pressure will decrease.
3. Conversely, if we decrease the volume of the container, then the particles will collide more often, and the pressure will increase.
4. If we double the volume of the container, we expect to get half the pressure (since there's twice as much space for the particles to fill, then on average, half as many particles will collide against a given area in a given amount of time. The particles will exert half as much force over a given area in a given amount of time).
5. If we halve the volume of the container, we expect to get double the pressure (since there's half as much space for the particles to fill, then on average, twice as many particles will collide against a given area in a given amount of time. The particles will exert twice as much force over a given area in a given amount of time).
6. Volume and pressure are therefore inversely proportional to one another.
  1. This means that volume is proportional to 1 ÷ Pressure.
  2. Therefore, volume is equal to some constant (k) × 1 ÷ Pressure, or k ÷ Pressure.
  3. Therefore, Volume x Pressure = some constant (k). PV = k.
Gay-Lussac's Law states that if there is a fixed volume, then the pressure will be directly proportional to the temperature.
1. If particles have more thermal energy, they will have more kinetic energy, and they will move faster.
2. If particles move faster, then they will collide with the walls of the container more often, and more particles will collide against a given area in a given amount of time. The particles will exert more force over a given area in a given amount of time. Thus, the pressure will be greater if the particles move faster.
3. Conversely, if we decrease the temperature, then the particles will move slower and collide against a given area less often. The particles will exert less force over a given area in a given amount of time, and the pressure will decrease.
4. If we double the temperature, we expect to get double the pressure (since the particles will be moving twice as fast, they'll collide against a given area twice as much in a given amount of time. They'll exert twice as much force over a given area in a given amount of time).
5. If we halve the temperature, we expect to get half the pressure (since the particles will be moving half as fast, they'll collide against a given area half as much in a given amount of time. They'll exert half as much force over a given area in a given amount of time).
6. Pressure and temperature are therefore directly proportional to each other.
  1. This means that pressure is equal to some constant × temperature.
  2. P = kT
Charles' Law states that if there is a fixed pressure, then the volume of the gas will be directly proportional to its temperature.
1. Following the two previous laws mentioned above, if we keep the pressure fixed but increase the volume, the particles will have to move faster to collide against the walls of the container at the same rate as before. They particles will have to move faster to exert the same amount of force over a given area in a given amount of time. The temperature will have to increase.
2. Conversely, if the pressure remains fixed but the volume decreases, the particles will have to move slower to collide against the walls of the container at the same rate as before. They particles will have to move more slowly to exert the same amount of force over a given area in a given amount of time. The temperature will have to decrease.
3. If we double the volume, we need to double the temperature to maintain the same pressure (the particles will need to move, on average, twice as fast to cover the extra space and exert the same amount of force over a given area in a given amount of time as they did before).
4. If we halve the volume, we will need to halve the temperature as well to maintain the same pressure (the particles will need to move, on average, half as fast to cover the extra space and exert the same amount of force over a given area in a given amount of time as they did before).
5. Volume and temperature are therefore directly proportional:
  1. This means that volume is equal to some constant times temperature.
  2. V = kT

So far, combining these three laws gives us PV = kT, where k is some constant

Avogadro's Law states that if the pressure and temperature for two gases are constant, then equal volumes of those two gases means that there are an equal number of particles. While this subpage won't try to prove Avogadro's law, this law is necessary to finishing the ideal gas law:
1. If we have two gases, both with constant and equal pressures and temperatures, and both gases have a volume of V₁, then both gases have the same number of particles according to Avogadro's Law.
2. If we increase the volume of both gases to V₂, while maintaining the same pressure and temperature for both gases, then we need to add more particles to both gases. In accordance with Avogadro's Law, the number of particles that both gases now have is, once again, equal, as both gases have the same pressure, temperature, and volume.
3. Both gases saw no increase in their pressure or temperature. Both gases also saw the same increase in their volume, and the same increase in the number of particles they had.
4. Under a fixed pressure and temperature, an increase in the number of particles in a gas will result in some increase in the volume of the gas. If we increase the number of particles in a gas, we will see an increase in the volume of the gas.
5. Similarly, if we decrease the number of particles in a gas, we will see a decrease in the volume of the gas.
6. If we double the number of particles while keeping the pressure and temperature constant, the volume of the gas will accordingly double.
7. Similarly, if we halve the number of particles while keeping the pressure and temperature constant, the volume of the gas will accordingly halve.
8. Thus, under a fixed temperature and pressure, the number of particles in a gas is proportional to its volume.
  1. Volume is equal to some constant times the number of particles.
  2. V = kn (where n is the number of particles in moles, and k is a constant)

Combining the four equations we have so far gives us PV = knT. The value of k in this equation turns out to be R, the Universal Gas Constant. This gives us the equation of the ideal gas law:

PV = nRT

Given three of the variables in this equation for a gas $R is a constant, so we don't consider R to be a variable$ , we can calculate the value of the fourth variable for that gas. And since Avogadro's Law shows that this is the same for any two gases, we know that not only does R remain the same for all gases, but we also know that the ideal gas law applies to all gases, in all instances.

Well, almost all instances.

Exceptions to the Ideal Gas Law

There are two major exceptions to using the ideal gas law - two instances where it doesn't work:

When the pressure of the gas exceeds a certain level. The ideal gas law assumes that the particles of the gas have no volume. Most of the time, this is practically true for real gases as well - the particles do have volume, but it is so small compared to the space between them that the volume of the particles is practically negligible. However, when the pressure gets too high, the volumes of the particles start to become noticeable.

When the pressure gets to that level, the volumes of the particles start to become noticeable, but the ideal gas law equation isn't accounting for that. So in reality, the gas takes up more space than predicted by the equation, since the volumes of the particles is noticeable enough to be added to and change the total volume of the gas $and the equation assumes the volumes of the particles to be 0$ .
When the temperature of the gas is too low. The ideal gas law assumes that there are no intermolecular forces between the gases. When the temperature of the gas is too low, however, the molecules move slowly enough that intermolecular forces of attraction noticeably form between the molecules, bringing them closer together.

When the temperatures drops to that level, the ideal gas law predicts that the gas will take up a lower volume. However, when the temperature is that low, the intermolecular forces between the gas particles start to pull strongly on the particles, bringing them even closer together. The ideal gas law doesn't account for this, since it assumes that there are no intermolecular forces between the gas particles. Therefore, the intermolecular forces bring the particles closer together than predicted by the ideal gas law, and the gas ends up with a smaller volume than predicted by the ideal gas law.