# 18.25 An Ideal Gas

A gas will fill all the space available in a sealed container; it has no surfaces except where it meets the walls of its container (post 16.37). Let’s think about a gas in the cylinder (coloured yellow in the picture above) that is sealed by a piston (red) that can move up and down in the cylinder (black). Applying a force (post 16.13) increases the pressure acting on the gas. This pressure is equal to the total force acting (the applied force minus the frictional force, post 16.19, between the piston and the cylinder) divided by the cross-sectional area of the cylinder (post 17.5). Now let’s keep our sample of gas at a constant temperature (post 16.34). For most gases at the pressures that can be generated easily, doubling the pressure then halves the volume; if the pressure is increased three times, the volume is reduced to a third of its initial value. We say that the volume is inversely proportional to the pressure. This result van be expressed mathematically by

p = C/V                    (1)

where p represents pressure, V represents volume and C is a constant for a given mass of gas at a given temperature.

The result expressed in equation 1 is called Boyle’s law. It is often written in the form

pV = C                    (2)

which is obtained from equation 1 by multiplying both sides by V.

Now let’s suppose that we keep p constant (by keeping the applied force constant) and increase the temperature T. If T is measured on the Kelvin scale (post 16.34), doubling the temperature doubles the volume, tripling the temperature triples the volume, and so on. This applies to most gases at pressures that can be generated easily. We say that V is proportional to T and can write this result as

V = C’T                      (3)

where C’ is a constant for a given mass of gas at a given pressure. This result is often called Charles’ law.

Finally, let’s suppose that we keep V constant by changing the applied force whenever the temperature changes, to keep the piston in a fixed position. We then find, for most gases at pressures that can be easily generated that

P = C’’T                     (4)

where C’’ is a constant for a given mass of gas at a given volume.

Equations 2, 3 and 4 are special cases of the general result that

pV = KT                    (5)

where K is a constant for a given mass of gas. (For example, when T is constant, KT is a constant; if we call this constant C, we get equation 2.)

Now let’s suppose that we have 1 mole (post 17.48) of gas in the cylinder. It turns out, when equation 5 works reasonably well, that K is then a constant for all gases. We call the resulting constant the ideal gas constant (or the gas constant), R (whose value is 8.314 J.mol-1.K-1), so that

pV = RT.                    (6)

If we have n moles of gas in our sample, the previous equation becomes

pV = nRT.                    (7)

Equation 7 is called the ideal gas equation.

Any gas whose behaviour can be described by equation 7 is called an ideal gas?

Why does the ideal gas equation work for most gases most of the time? In post 16.37, we saw that a gas contains molecules that are free to move anywhere. Molecules that contain a small number of atoms are so small (post 16.27) that their volume is usually negligible compared to the total volume of the gas. So, the probability that two molecules will collide with each other is negligible. So, when we halve the volume of the gas, the molecules are twice as likely to collide with the walls. Each collision involves a change in the momentum of the molecule involved (post 17.30), so it exerts a force on the wall of the container (post 16.13). When we divide the total force of all the molecular collisions by the surface area of the container, we have calculated the pressure of the gas (post 17.5). The system in the picture, is in equilibrium when the gas pressure equals to applied pressure, because the total force acting on the piston is zero. So halving the volume doubles the pressure.

Now we’ll look at a slightly different way of writing the ideal gas equation. Equation 6 applies to 1 mole of gas. We have seen, in post, that 1 mole contains N molecules, where N is Avogadro’s number (post 17.48). So, if we want to think about the behaviour of molecules, we can write equation 6 in the form

pV = N(R/N)T.                    (8)

Since both R and N are constants, we define another constant k = R/N. Equation 8 can now be written as

pV = NkT.                    (9)

This is another common form of the ideal gas equation; k is called Boltzmann’s constant and has the value 1.3806 J.K-1. We could use the statistical definition of entropy, in post 16.35, as an alternative definition of k; the two definitions are equivalent but it would need a lot of background information to explain.

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