The Gas Laws

 

The first “gas law’ was formulated by Robert Boyle in 1662. He related the pressure and volume of a gas in a closed container through his gas law,  His experiments were carried out at room temperature, and the law is valid only if the temperature is held constant during the process. We can call the constant C₁. Almost a century after Boyle’s work, Jacques Charles and, independently, Louis Gay-Lussac held the pressure of gasses constant and studied the relationship between temperature and volume. Their gas law may be written in the form  A graph of Boyle’s law for two different temperatures, and Charles’ law for a particular pressure is shown in the diagram. In the plot of Boyle’s law, T₂ is greater than T₁. The straight line for Charles’ law is independent of the type of gas (unlike the situation where pressure is plotted as a function of temperature, where the lines for different gasses give different slopes - but all converge at -273.16 ^oC). This relation also approaches the temperature -273.16 ^oC as the volume approaches zero.

 

So, we have two relations involving p, V, and T (and a third where V is held constant, if you wish, but two is enough for our purposes). Boyle’s law can be written as  and Charles’ law as , the first with T held constant and the second with p held constant. Look at the graph of Boyle’s law: A sample of gas starts with pressure p₁, temperature T₁ and volume V₁ and later finds itself at p₃, V₃ with the temperature held at T₁ (i.e. the system parameters move up the T₁ = constant curve). We can then write . Next our system moves to  while holding the pressure constant (Charles’ law). For this case we write  Eliminating V₃ from these and since  we arrive at the more general form of the combined gas laws as  or, equally,  Can we find a value for the constant? Remember that these laws were arrived at experimentally, and further experiments showed that the constant is directly proportional to the mass (m) of gas used, and inversely proportional to the molecular weight (M) of the gas. But m is itself directly proportional to M (  where n is the number of moles of the gas) and so the ratio of these is a constant*. We thus write (moving the T to the right hand side, in keeping with tradition),

 

This is the ideal gas law, with R the universal gas constant equal to 8.31434. It’s called ideal because it is valid for only so-called ideal gasses. At the time these laws were formulated this meant dilute gasses. As the pressures of the gasses involved were increased, the relation deteriorated. At atmospheric pressures and “ordinary” temperatures, gasses are essentially ideal. What are the criteria for an ideal gas? When treating the gas as a fluid, the answer is “dilute enough”. When treating the gas as a collection of molecules in constant motion the answer becomes much more satisfying, and leads to the kinetic theory of gasses.