# 6.7: Gas Law Equations: Relating the Pressure, Volume, and Temperature of a Gas

Learning Objectives
• Write an equation for the Combined Gas Law.

After completing the two-variable experiments that are described in the previous three sections of this chapter, scientists noticed that, if present, the variables for the pressure and the volume of a constant amount of gas were always written in the numerator of the equations that had been derived.  Furthermore, if applicable, the variable for temperature was always incorporated into the denominator of those equations.  The Combined Gas Law, which is shown below, was developed to summarize these collective observations.  This equation can also be stated without explicitly writing the multiplication symbols and by using variables that have abbreviated or modified subscripts.

$$\dfrac{\rm{P_{initial}} × \rm{V_{initial}}}{\rm{T_{initial}}}$$ = $$\dfrac{\rm{P_{final}} × \rm{V_{final}}}{\rm{T_{final}}}$$

$$\dfrac{\rm{P_{i}} \rm{V_{i}}}{\rm{T_{i}}}$$ = $$\dfrac{\rm{P_{f}} \rm{V_{f}}}{\rm{T_{f}}}$$

$$\dfrac{\rm{P_{1}} \rm{V_{1}}}{\rm{T_{1}}}$$ = $$\dfrac{\rm{P_{2}} \rm{V_{2}}}{\rm{T_{2}}}$$

The variables that are present in this equation are related by both multiplication and division.  As stated in Section 6.3, two directly-proportional quantities are quantitatively-related through division, and two indirectly-proportional values are mathematically-associated through multiplication.  However, these qualitative relationships are only applicable to systems that involve two variables.  Therefore, since three variables are incorporated into the Combined Gas Law, its collective quantities are neither directly nor indirectly proportional.