The combined gas law combines the relationships between pressure, volume, and temperature of a gas. It applies when the amount of gas and the number of gas particles remain constant. The equation for the combined gas law is as follows:

(P1 * V1) / T1 = (P2 * V2) / T2

where P1, V1, and T1 represent the initial pressure, volume, and temperature, and P2, V2, and T2 represent the final pressure, volume, and temperature, respectively.

The combined gas law allows us to predict how changes in one variable will affect the other variables. For example, if we decrease the volume of a gas sample while keeping the pressure constant, the temperature must also decrease in order to maintain the equality in the equation. This can be explained using the kinetic theory of gases, which states that the average kinetic energy of gas particles is directly proportional to the temperature.

On the other hand, the ideal gas law combines the relationships between pressure, volume, temperature, and the number of moles of gas. It is expressed as:

P * V = n * R * T

where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. The ideal gas law allows us to calculate the missing variable when the other variables are known.

For example, if we have a sample of gas with a known pressure, temperature, and volume, we can use the ideal gas law to calculate the number of moles of gas present. This is useful in various applications, such as determining the amount of reactants needed in a chemical reaction.

In summary, the combined gas law and ideal gas law provide mathematical relationships that help us understand how pressure, volume, temperature, and the number of moles of gas are related. These laws are essential in studying the behavior of gases and have practical applications in fields such as chemistry, physics, and engineering.