Grokking the Thermodynamic Theory of Gases

©Fernando Caracena2014

The Various Gas Laws

A gas is a fluid which expands or contracts to fill a closed container. Gases exert a force on the walls of the container through a property called pressure, which is an amount of force exerted per unit square of area perpendicular to the walls. The units of pressure are therefore those of force per square length. For example, in SI units, pressure is measured in units of Newtons per square meter (N m-2), or in units called Pascals (Pa). At the filling station, our tire pressure is measured in pounds per square inch (psi) in the U S.

When a gas is compressed, it responds by a combination of increasing in pressure and temperature. If a gas is heated, it will respond by increasing in pressure, and if it can, in volume as well.

Figure 1: Gas under constant pressure:
A block of weight, m*g, exerts a constatnt, downward force on a piston, which results in a constant pressure. Assume that the seal on the piston is tight enoungh to avoid leaks, but frictionless enough to allow the piston to move up and down freely.

Consider a gas enclosed in such a way that it is under constant pressure (Fig.1).Such a gas will expend when heated, or contract when cooled. More specifically, the volume will vary linearly with temperature. This of course happens as long as the enclosed substance remains a gas, which happens within limits of temperature, pressure and volume, which are defined by phase transitions. Cooled enough, the gas may begin to condense into a liquid, or even directly into a solid, in which case gas laws no longer apply.

Summary of Gas Laws

Here is a summary of all results (in the form of equations) that you can obtain for the behavior of a gas under constant pressure, but variable temperature and volume:

P= P0, (1)

V= m* T +b, (2a)

where P0 is a constant.

The Absolute Temperature Scale

Scientists working with gases found it convenient to invent a new temperature scale by absorbing the constant b in (2) as part of the redefinition of T

Ta = T+T0

so that

b = m* T0.

The value

T0 = 273.16

applies to the conversion from Celsius to absolute units, the Kelvin.

From here on, we shall use the Kelvin temperature scale in which the relation of the volume of the gas to its temperature, under constant pressure, is linear,

V= m* T. (2b)

Note that at T=0, the volume of the gas would go to zero, but before that would happen the gas would undergo phase transitions and become liquid or solid. Also, the gas law described in (2b) is known as Charles's Law.

Robert Boyle confirmed and published a discovery made by two other scientists that a gas held at a fixed temperature but of variable volume and pressure responds in such a way that the pressure and volume are inversely proportional,

T=const1. (3a)

P V = const2. (3b)

This relationship is known as Boyle's Law.

The Ideal Gas Law

All possible combinations of the above experiments are summed up in a simple equation,

P V = K T, (4)

which is called the Ideal Gas Law. Note that (4) is an empirical relation that packs a lot of information about the behavior of gases in a deceptively simple form. This form represents a powerful tool that we can use to grok the kinetic theory of gases.

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