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

Learning Objectives
• Write an equation for the Ideal Gas Law.
• State the value of the Ideal Gas Constant.

As stated in Section 6.7, if present, the variables for the pressure and the volume of gas are always written in the numerator of the Gas Law equations, and the variable for temperature is always incorporated into the denominator of those equations.  The development of Avogadro's Law established that the variable that corresponds to the amount of gas that is present should also be written in the denominator of a Gas Law equation.  The equation that is shown below was developed to incorporate this additional observation.

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

Recall that each of the Gas Law relationships was initially expressed by equating the product or quotient of two of the measurable properties of gases to a constant, k.  Incorporating this constant into the relationship that is shown above results in the derivation of the following equation sequence.

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

Because all of the values in an equation sequence are equal to one another, the value of this constant can be independently-related to either the left- or right-most quantity that is shown above.  Therefore, the references to the "initial" and "final" states of the gas can be removed, resulting in the equation that is shown below.

$$\dfrac{\rm{P} × \rm{V}}{\rm{T} × \rm{n}}$$ = $$\rm{k}$$

Recall that the values of the constants in Boyle's LawGay-Lussac's LawCharles's Law, and Avogadro's Law are not universal constants.  However, because the equation that is shown above relates all four of the measurable quantities of gases, scientists determined that the value of its corresponding constant, k, is universally-applicable to all gases, under certain circumstances.  In order for k to be classified as a universal constant, the volume of a gas's constituent particles must be negligible, relative to the total volume that is occupied by the gas, and, furthermore, those individual particles cannot interact in any way.  When those conditions are satisfied, the associated constant is represented as R, which has an approximate value of 0.08206 atm⋅L/K⋅mol.  This unit consists of both a numerator and a denominator, as indicated by the "/", and the "·"s denote the multiplicative relationships between the pressure and volume units and between the temperature and amount units.  The equation that is shown below, which is known as the Ideal Gas Law, incorporates the variable for the Ideal Gas Constant, R.

$$\dfrac{\rm{P} × \rm{V}}{\rm{T} × \rm{n}}$$ = $$\rm{R}$$

Finally, the mathematical process of cross-multiplication can be used to eliminate the denominator portion of the fraction that is shown above.  When cross-multiplying, the numerator on one side of an equation is multiplied by the denominator on the other side of the equal sign.  The process is repeated for the remaining quantities, and the resultant products are equated to one another.  This alternative representation of the Ideal Gas Law, which is stated without explicitly writing the multiplication symbols, is shown below.

PV = nRT

Finally, as stated above, in order for k to be classified as a universal constant, the particles of the gas that is being studied must not interact with one another and must have negligible volumes.  However, only ideal, or perfect, gases can be described using these parameters.  As a result, the measurable properties of real gases can only be approximated using the Ideal Gas Law.