The basic gas laws

 

Contents of this page   Boyle's Law The law of Charles and Gay-Lussac Avogadro's Law The Ideal Gas Equation of State The P-V-T surface Summary

The "pneumatic" era of chemistry began with the discovery of the vacuum around 1650 which clearly established that gases are a form of matter. The ease with which gases could be studied soon led to the discovery of numerous emprical (experimental) laws that proved fundamental to the later development of chemistry and led indirectly to the atomic view of matter.

 

 

 

Robert Boyle (1627-91) showed that the volume of air trapped by a liquid in the closed short limb of a J-shaped tube decreased in exact proportion to the pressure produced by the liquid in the long part of the tube. The trapped air acted much like a spring, exerting a force opposing its compression. Boyle called this effect “the spring of the air", and published his results in a pamphlet of that title.

 

volume pressure P × V 96.0 2.00 192 76.0 2.54 193 46.0 4.20 193 26.0 7.40 193 Some of Boyle's actual data are reproduced above.

The effect can be seen in a simple J-shaped tube in which air is trapped in the short limb as mercury is poured into the right side. The difference between the heights of the two mercury columns gives the pressure (76 cm = 1 atm), and the volume of the air is calculated from the length of the air column and the tubing diameter. In Boyle's experiment he used a simple air pump invented by his friend Robert Hooke.

 

Boyle's law can be expressed as

PV = constant

or, equivalently,

P₁V₁ = P₂V₂

These relations hold true only if the number of molecules n and the temperature are constant. This is a relation of inverse proportionality; any change in the pressure is exactly compensated by an opposing change in the volume. As the pressure decreases toward zero, the volume will increase without limit. Conversely, as the pressure is increased, the volume decreases, but can never reach zero. There will be a separate P-V plot for each temperature; a single P-V plot is therefore called an isotherm.

 

Shown here are some isotherms for one mole of an ideal gas at several different temperatures. Each plot has the shape of a hyperbola— the locus of all points having the property x y = a, where a is a constant. You will see later how the value of this constant (PV=25 for the 300K isotherm shown here) is determined. It is very important that you understand this kind of plot which governs any relationship of inverse proportionality. You should be able to sketch out such a plot when given the value of any one (x,y) pair.

 

A related type of plot with which you should be familiar shows the product PV as a function of the pressure. You should understand why this yields a straight line, and how this set of plots relates to the one immediately above. Try this interactive Boyle's Law experiment from Davidson College

 

 

 

 

 

 

 

All matter expands when heated, but gases are special in that their degree of expansion is independent of their composition. The French scientists Jacques Charles (1746-1823) and Joseph Gay-Lussac (1778-1850) independently found that if the pressure is held constant, the volume of any gas changes by the same fractional amount (1/273 of its value) for each C° change in temperature.

 

A graphical expression of the law of Charles and Gay-Lussac can be seen in these plots of the volume of one mole of an ideal gas as a function of its temperature at various constant pressures.

• What do these plots show? The straight-line plots show that the ratio V/T (and thus dV/dT) is a constant at any given pressure. Thus we can express the law algebraically as      V/T = constant   or   V₁/T₁ = V₂/T₂
• What is the significance of the extrapolation to zero volume? If a gas contracts by 1/273 of its volume for each degree of cooling, it should contract to zero volume at a temperature of –273°C. This, of course, is the absolute zero of temperature, and this extrapolation of Charles' law is the first evidence of the special significance of this temperature.
• Why do the plots for different pressures have different slopes? The lower the pressure, the greater the volume (Boyle's law), so at low pressures the fraction (V/273) will have a larger value. You might say that the gas must "contract faster" to reach zero volume when its starting volume is larger.

 

Try this interactive exploration of the Law of Charles and Gay-Lussac

Carleton University lecture demonstration video of Charles' Law (YouTube)

 

 

 

 

 

 

Gay-Lussac noticed that when two gases react, they do so in volume ratios that can always be expressed as small whole numbers. Thus when hydrogen burns in oxygen, the volume of hydrogen consumed is always exactly twice the volume of oxygen. The Italian scientist Amedeo Avogadro (1776-1856) drew the crucial conclusion: these volume ratios must be related to the relative numbers of molecules that react, and thus the famous "E.V.E.N principle":

Equal volumes of gases, measured at the same temperature and pressure, contain equal numbers of molecules.

Avogadro's law thus predicts a directly proportional relation between the number of moles of a gas and its volume.

This relationship, originally known as Avogadro's Hypothesis, was crucial in establishing the formulas of simple molecules at a time (around 1811) when the distinction between atoms and molecules was not clearly understood. In particular, the existence of diatomic molecules of elements such as H₂, O₂, and Cl₂ was not recognized until the results of combining-volume experiments such as those depicted below could be interpreted in terms of the E.V.E.N. principle.

 

Once it was shown that equal volumes of hydrogen and oxygen do not combine in the manner depicted in (1), it became clear that these elements exist as diatomic molecules and that the formula of water must be H₂O rather than HO as previously thought.

 

 

 

 

If the variables P, V, T and n (the number of moles) have known values, then a gas is said to be in a definite state, meaning that all other physical properties of the gas are also defined. The relation between these state variables is known as an equation of state. By combining the expressions of Boyle's, Charles', and Avogadro's laws (you should be able to do this!) we can write the very important ideal gas equation of state

where the proportionality constant R is known as the gas constant. This is one of the few equations you must commit to memory in this course; you should also know the common value and units of R.

We will see later that all real gases behave more and more like an ideal gas as the pressure approaches zero. A pressure of only 1 atm is sufficiently close to zero to make this relation useful for most gases at this pressure.

In order to depict the relations between the three variables P, V and T we need a three-dimensional graph.

PVT surface for an ideal gas   Each point on the curved surface represents a possible combination of (P,V,T) for an arbitrary quantity of an ideal gas. The three sets of lines inscribed on the surface correspond to states in which one of these three variables is held constant. The red curved lines, being lines of constant temperature, or isotherms, are plots of Boyle's law. These isotherms are also seen projected onto the P-V plane at the top right. The yellow lines are isochors and represent changes of the pressure with temperature at constant volume. The green lines, known as isobars, and projected onto the V-T plane at the bottom, show how the volumes contract to zero as the absolute temperature approaches zero, in accordance with the law of Charles and Gay-Lussac.

 

 

 

 

 

 

 

This animated PVT surface was found at a now-defunct Japanese Web site.