Ideal Gas Law: Definition, Formula & Examples

One of the most fundamental laws in thermodynamics is the ideal gas law, which allows scientists to predict the behavior of gases that meet certain criteria.

Simply speaking, an ideal gas is a theoretically perfect gas that makes the math easier. But what math? Well, consider that a gas is made up of an incredibly large number of atoms or molecules all free to move past each other.

A container of gas is like a container of thousands upon thousands of tiny balls all jostling around and bouncing off of each other. And sure, it's easy enough to study the collision of just two such particles, but to keep track of every single one of them is virtually impossible. So if each gas molecule is acting like an independent particle, how can you understand the workings of the gas as a whole?

The kinetic theory of gases provides a framework for understanding how gas behaves. As described in the previous section, you can treat a gas as a collection of a large number of extremely small particles undergoing constant rapid motion.

Kinetic theory treats this motion as random since it is the result of multiple rapid collisions, making it too difficult to predict. It is by treating this motion as random and using statistical mechanics that an explanation for the macroscopic properties of a gas can be derived.

It turns out that you can describe a gas pretty well with a set of macroscopic variables instead of keeping track of each molecule on its own. These macroscopic variables include temperature, pressure and volume.

How these so-called ​state variables​ relate to each other depends on the properties of the gas.

State variables are quantities that describe the state of a complex dynamical system, such as a gas. Gases are often described by state variables such as pressure, volume and temperature.

Pressure is defined as the force per unit area. The pressure of a gas is the force per unit area it exerts on its container. This force is a result of all of the microscopic collisions occurring within the gas. As the gas molecules bounce off the sides of the container, they exert a force. The greater the average kinetic energy per molecule, and the greater the number of molecules in a given space, the greater the pressure will be. The SI units of pressure are newtons per meter, or pascals.

Temperature is a measure of the average kinetic energy per molecule. If all of the gas molecules are thought of as small points jostling around, then the temperature of the gas is the average kinetic energy of those small points.

A higher temperature corresponds to more rapid random motion, and a lower temperature corresponds to slower motion. The SI unit of temperature is the Kelvin, where absolute zero Kelvin is the temperature at which all motion ceases. 273.15 K is equal to zero degrees Celsius.

The volume of the gas is a measure of the space occupied. It is simply the size of the container the gas is confined within, measured in cubic meters.

These state variables arise from the kinetic theory of gases, which allows you to apply statistics to the motion of the molecules and derive these quantities from things such as the root mean square velocity of the molecules and so on.

An ideal gas is a gas for which you can make certain simplifying assumptions that allow for easier understanding and calculations.

In an ideal gas, you treat the gas molecules as point particles interacting in perfectly elastic collisions. You also assume that they are all relatively far apart and that intermolecular forces can be ignored.

At standard temperature and pressure (stp) most real gases behave ideally, and in general gases are most ideal at high temperatures and low pressures. Once the assumption of "idealness" is made, you can start looking at the relationships between pressure, volume and temperature, as described in the following sections. These relationships will eventually lead to the ideal gas law itself.

Boyle's law states that at constant temperature and amount of gas, pressure is inversely proportional to volume. Mathematically this is represented as:

\(P_1V_1=P_2V_2\)

Where ​P​ is pressure, ​V​ is volume and the subscripts indicate initial and final values.

If you think about kinetic theory and the definition of these state variables for a moment, it makes sense why this law should hold. The pressure is the amount of force per unit area on the walls of the container. It depends on the average energy per molecule, since the molecules are colliding with the container, and how densely packed these molecules are.

It seems reasonable to assume that if the volume of the container becomes smaller while temperature remains constant, then the total force exerted by the molecules should remain the same, since they same in number and same in energy. However, since pressure is force per unit area and the surface area of the container has shrunk, then the pressure should increase accordingly.

You may have even witnessed this law in your everyday life. Have you ever noticed that a partially inflated helium balloon or a bag of potato chips seems to expand/inflate considerably when you go up in elevation? This is because, even though the temperature might not have changed, the air pressure outside decreased, and hence the balloon or the bag was able to expand until the pressure inside was the same as the pressure outside. This lower pressure corresponded to a higher volume.

Charles' law states that, at constant pressure, volume is directly proportional to temperature. Mathematically, this is:

\(\frac{V_1}{T_1}=\frac{V_2}{T_2}\)

Where ​V​ is volume and ​T​ is temperature.

Again, if you consider kinetic theory, this is a reasonable relationship. It basically states that a decrease in volume would correspond to a decrease in temperature if pressure is to remain constant. Pressure is force per unit area, and decreasing the volume decreases the container surface area, so in order for the pressure to remain the same when volume is decreased, the total force also has to decrease. This would only happen if the molecules have a lower kinetic energy, meaning a lower temperature.

This law stats that, at constant volume, pressure is directly proportional to temperature. Or mathematically:

\(\frac{P_1}{T_1}=\frac{P_2}{T_2}\)

Since pressure is force per unit area, if the area stays constant, the only way for the force to increase is if the molecules move faster and collide harder with the surface of the container. So, the temperature increases.

Combining the three previous laws yields the ideal gas law via the following derivation. Consider that Boyle's law is equivalent to the statement ​PV​ = constant, Charles' law is equivalent to the statement ​V/T​ = constant and Guy-Lussac's law is equivalent to the statement ​P/T​ = constant. Taking the product of the three relationships then gives:

\(PV\frac{V}{T}\frac{P}{T} = \frac{P^2V^2}{T^2} = \text{constant}\)

Or:

\(PV=\text{constant}\times T\)

The value of the constant, not surprisingly, depends on the number of molecules in the gas sample. It can be expressed as either constant = ​nR​ where ​n​ is the number of moles and ​R​ is the universal gas constant (​R​ = 8.3145 J/mol K), or as constant = ​Nk​ where ​N​ is the number of molecules and ​k​ is Boltzmann's constant (k = 1.38066 × 10^-23 J/K). Hence the final version of the ideal gas law is expressed:

\(PV = nRT = NkT\)

This relationship is an equation of state.

​Example 1:​ A large, helium-filled balloon is being used to lift scientific equipment to a higher altitude. At sea level, the temperature is 20 C and at the higher altitude the temperature is -40 C. If the volume changes by a factor of 10 as it rises, what is its pressure at the higher altitude? Assume the pressure at sea level is 101,325 Pa.

​Solution:​ The ideal gas law, slightly rewritten, can be interpreted as ​PV/T​ = constant, or:

\(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\)

Solving for ​_P₂_​, we get the expression:

\(P_2 = \frac{P_1V_1T_2}{V_2T_1}\)

Before plugging in numbers, convert the temperatures to Kelvin, so ​_T₁​ = 273.15 + 20 = 293.15 K, ​T₂​ = 273.15 – 40 = 233.15 K. And while you haven't been given the exact volume, you do know that the ratio ​V₁/V₂_​ = 1/10. So the final result is:

\(P_2 = \frac{101,325\times 233.15}{10\times 293.15} = 8,059 \text{ Pa}\)

​Example 2:​ Find the number of moles in 1 m³ of gas at 300 K and under 5 × 10⁷ Pa of pressure.

​Solution:​ Rearranging the ideal gas law, you can solve for ​n​, the number of moles:

\(n = \frac{PV}{RT}\)

Plugging in numbers then gives:

\(n = \frac{5\times 10^7\times 1}{8.3145\times 300} = 20,045 \text{ moles}\)

Avogadro's law states that gases at equal volumes, pressures and temperatures necessarily have the same number of molecules. This follows directly from the ideal gas law.

If you solve the ideal gas law for the number of molecules, as was done in one of the examples, you get:

\(n = \frac{PV}{RT}\)

So if everything on the right-hand side is held constant, there is only one possible value for ​n​. Note that this is of particular interest because it holds true for any type of ideal gas. You can have two different gases, but if they are at the same volume, pressure and temperature, they contain the same number of molecules.

Of course there are many instances in which real gases do not behave ideally. Recall some of the assumptions of an ideal gas. The molecules must be able to be approximated as point particles, taking up essentially no space, and there must not be any intermolecular forces at play.

Well, if a gas is compressed enough (high pressure), then the size of the molecules comes into play, and the interactions between molecules becomes more significant. At extremely low temperatures as well, the energy of the molecules might not be high enough to cause a roughly uniform density throughout the gas either.

A formula called the Van der Waals equation helps correct for a particular gas's deviation from ideal. This equation can be expressed as:

\((P+\frac{an^2}{V^2})(V-nb) = nRT\)

This is the ideal gas law with a correction factor added to ​P​ and another correction factor added to ​V​. The constant ​a​ is a measure of the strength of attraction between molecules, and ​b​ is a measure of the size of the molecules. At low pressures, the correction in the pressure term is more important, and at high pressures the correction in the volume term is more important.