# What is the general gas law?

The **general gas law** , sometimes also called the combined gas law, is the combination of Boyle’s, Charles’s, and Gay-Lussac’s laws. It is a law that describes the relationship between pressure, temperature, and volume of a fixed quantity of an ideal gas.

The statement of this law says:

*“If the number of moles of a gas is kept constant, the relationship between the Pressure-Volume product and the temperature remains constant.”*

In mathematical form, the general gas law is expressed like this:

where P represents the pressure of the gas, V its volume, T its absolute temperature, and K is a constant of proportionality, the value of which depends on both the amount of gas present and the units in which the other variables are expressed.

**Alternative forms of the general gas law**

**As a law of proportionality**

An alternative way of stating the general gas law is in the form of a proportionality law:

*“For any fixed quantity of a gas, the product of its pressure and its volume is directly proportional to the temperature.”*

This is equivalent to multiplying both sides of the first equation by the temperature.

**Relationship between initial and final state**

Like the Boyle, Charles, and Gay-Lussac laws, the general gas law can be expressed as a relationship between the initial state and the final state of a gas undergoing a state change.

Unlike the previous laws, none of the three variables need to remain constant, just the number of moles. That is, the PxV / T relationship will be the same in the initial state, *i,* and in the final state, *f* . In other words, the general gas law can also be expressed mathematically as:

**Derivation of the formula for the general gas law**

As mentioned above, the general gas law comes from the combination of the Boyle, Charles, and Gay-Lussac laws. These laws are presented below:

If from each of these laws we solve for the constant *k* , and then multiply them with each other, we obtain:

Now, taking the square root of both members, the general gas law is obtained:

**The general gas law vs. the ideal gas law**

The general gas law should not be confused with the ideal gas law. Despite being two very similar and also closely related laws, the general gas law combines only the laws of Boyle, Charles and Gay-Lussac.

For its part, the ideal gas law adds Avogadro’s principle, according to which ” *equal volumes of different gases, measured under the same conditions of temperature and pressure, contain the same number of particles* . *“*

In the following equations, the difference between these two laws can be more easily observed:

Note that the main difference between both laws is that the ideal gas law includes *n* , which represents the number of moles, and also instead of the constant *K, it has* the constant R which is the constant of the ideal gases.

We could say that the ideal gas law is more general than the general gas law, since it can be applied to any quantity of a gas under any set of pressure, temperature and volume conditions. Instead, the application of the general gas law requires that the quantity of the gas remain constant.

**Examples of the application of the general gas law**

Here are some examples of typical problems in which the general gas law can be applied:

**Example 1: An air bubble under water**

Suppose that a diver who is at a depth of 20 m underwater where the pressure is 3.00 atm and the temperature is 15 ° C releases a puff of air and one of the bubbles has an initial volume of 100 cm ^{3} .

Determine the volume of the air bubble upon reaching the surface that is under standard conditions of temperature and pressure, assuming that the amount of air inside the bubble does not change as it rises.

**Solution**

First, we must extract all the data from the statement. It is understood that there are two different states for gas, one initial and one final, so we separate the data into two groups. It is also necessary to convert the temperatures to absolute temperature:

As the problem specifies that there is no change in the amount of gas (air) then we can apply the general gas law, from which we can solve for the final volume:

Therefore, the air bubble reaches a volume of 310 cm ^{3} upon reaching the surface.

**Example 2: High pressure**

How much will the pressure of a sample of an ideal gas increase that is at a room temperature of 25 ° C and a pressure of 1.00 atm and is compressed to one thousandth of its volume while heated to 1800 ° C in a sealed container?

**Solution**

As before, we start by extracting the data:

Since it says that the process is carried out in a sealed container, then the amount of the gas does not change, so the general gas law can be applied: