# 6.4: Gas Law Equations: Relating the Pressure and Volume of a Gas

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An English scientist named Robert Boyle investigated how varying the pressure of a constant amount of gas impacted its corresponding volume under isothermal, or constant-temperature, conditions. As previously stated, the application of pressure causes gaseous particles to move closer to one another, which decreases the overall volume of the gas. Therefore, the pressure and volume of a gas are *indirectly*, or *inversely*, proportional to one another, and, as a result, multiplying these quantities yields a constant, k_{1}, as shown below.

\({\rm{P}}\) × \({\rm{V}}\) = \(\rm{k_{1}}\)

Boyle determined that k_{1} is not a universal constant, because its value varies based on the identity of the gas that is being investigated. Because hundreds of unique gases can be studied, relating these, or any, measurable quantities of gases through this type of "constant" is highly impractical. However, the numerical value of k_{1} *is *constant when performing trials with the *same *type of gas. Therefore, the value of k_{1} after the completion of an experiment must be identical to its value at the beginning of that experiment, as shown in the following equation sequence.

\({\rm{P_{initial}}}\) × \({\rm{V_{initial}}}\) = \(\rm{k_{1}}\) = \({\rm{P_{final}}}\) × \({\rm{V_{final}}}\)

Because all of the values in an equation sequence are equal to one another, the left- and right-most quantities can be directly related to one another, as shown below.

\({\rm{P_{initial}}}\) × \({\rm{V_{initial}}}\) = \({\rm{P_{final}}}\) × \({\rm{V_{final}}}\)

By eliminating k_{1} from this mathematical statement, the application of this equation is no longer dependent on the identity of the gas that is being studied, which greatly diversifies its utility, relative to the first two equations that were derived. Therefore, since the relationship between the pressure and the volume of *any *gas can be modeled using the final equation that is shown above, chemists selected this format for representing the first Gas Law. This equation, which is known as **Boyle's Law**, can also be stated without explicitly writing the multiplication symbols and by using variables that have abbreviated or modified subscripts, as shown below.

\({\rm{P_{i}}}\)\({\rm{V_{i}}}\) = \(\rm{P_{f}}\)\({\rm{V_{f}}}\)

\({\rm{P_{1}}}\)\({\rm{V_{1}}}\) = \(\rm{P_{2}}\)\({\rm{V_{2}}}\)