The Unified Relationship: A Guide to the Combined Gas Law

The Combined Gas Law is a powerful and convenient gas law that merges Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single, comprehensive expression. It describes the relationship between the pressure, volume, and absolute temperature of a fixed amount of gas. While the simpler gas laws describe the relationship between two variables while holding the third constant, the Combined Gas Law is used for situations where all three properties—pressure, volume, and temperature—are subject to change simultaneously. It provides a way to predict the final state of a gas when its initial state and two of its final properties are known.

This law is incredibly useful in chemistry and physics for analyzing real-world gas behavior where it's often not possible to keep one variable perfectly constant. For example, when a weather balloon rises through the atmosphere, it experiences a decrease in external pressure and a decrease in temperature, both of which affect its volume. The Combined Gas Law allows us to calculate this change. It is an essential tool for any student or professional who needs a robust way to solve gas problems under varying conditions. This calculator simplifies the process, allowing you to solve for any of the six variables in the equation instantly.

The Combined Gas Law Formula

The law states that the ratio of the product of pressure and volume to the absolute temperature of a gas is equal to a constant. When comparing the same substance under two different sets of conditions, this relationship can be written as:

(P₁V₁) / T₁ = (P₂V₂) / T₂

Where:

P₁ and V₁ are the initial pressure and volume.
T₁ is the initial absolute temperature (in Kelvin or Rankine).
P₂ and V₂ are the final pressure and volume.
T₂ is the final absolute temperature.

As with all gas laws involving temperature, you must use an absolute scale (Kelvin or Rankine) for the calculation to be correct. All other units for pressure and volume must simply be consistent on both sides of the equation.

Relationship to the Simpler Gas Laws

The Combined Gas Law beautifully demonstrates how the other simple gas laws are related. You can derive each one by holding one variable constant:

If the temperature is constant (T₁ = T₂), the temperatures cancel out, and the formula simplifies to P₁V₁ = P₂V₂, which is Boyle's Law.
If the pressure is constant (P₁ = P₂), the pressures cancel out, and the formula simplifies to V₁/T₁ = V₂/T₂, which is Charles's Law.
If the volume is constant (V₁ = V₂), the volumes cancel out, and the formula simplifies to P₁/T₁ = P₂/T₂, which is Gay-Lussac's Law.

Relationship to the Ideal Gas Law

The Combined Gas Law is also a special case of the Ideal Gas Law (PV = nRT). In the Combined Gas Law, the amount of gas (n, in moles) is assumed to be constant. Therefore, the term PV/T is equal to nR, which is a constant. This shows that P₁V₁/T₁ = nR and P₂V₂/T₂ = nR, leading directly to the Combined Gas Law equation.

Combined Gas Law Calculator