# 6.5: Gas Law Equations: Relating the Pressure and Temperature of a Gas

Learning Objectives
• State whether the pressure and temperature of a gas are directly or indirectly proportional.
• Write an equation for Gay-Lussac's Law.

Joseph Gay-Lussac, a French chemist, studied how the pressure of a constant amount of gas responded to changes in temperature under isovolumetric, or constant-volume, conditions.  By increasing the temperature of a gas, its constituent particles move more quickly and, therefore, collide more often with the surfaces of the container in which they are held.  Because the pressure that is exerted by a gas is caused by these impacts, increasing the frequency of these collisions in a rigid container increases the pressure of the gas.  Therefore, the pressure and temperature of a gas are directly, or linearly, proportional to one another, and dividing these quantities yields a constant, k2, as shown below.

$$\dfrac{ \text{P}}{\text{T}}$$ = $$\rm{k_2}$$

Like k1, k2 is not a universal constant, because its value also varies based on the identity of the gas that is being studied.  Therefore, the most useful representation of Gay-Lussac's Law directly relates the pressure and temperature of a gas at the beginning of an experiment to their corresponding values after the completion of that experiment.  This equation, which is shown below, can also be written using variables that have abbreviated or modified subscripts.

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

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

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