Peoples Physics Book Version-2

20.1. Thermodynamics http://www.ck12.org

Combined Ideal Gas Law

By combining any two of the empirical gas laws above with Avogadro’s Law, we can derive the following relation-

ship, called theIdeal Gas Lawand states that fornmolecules of any ideal gas,

PV=nkT [6]

In this case, the constantk(a combination of the constants above) can be empirically measured, since,because of

the addition of Avogadro’s Law, it is identical for all ideal gases. Note that the constants in the empirical gas

laws arenotnecessarily identical for all ideal gases (why?). Its value isk= 1. 38 × 10 −^23 J/K and it is known as the

Boltzmann Constant.

A different, though completely equivalent form of the ideal gas law is:

PV=NRT [ 7 ]

Vis the volume,Nis the number of moles of the gas (Ris the universal gas constant= 8 .315 J/K mol); this form is
often more useful for thermodynamics.
The relationship betweennandN above, that is the number of molecules per mole of a substance, is called
Avogadro’s number. By comparing the two equations above, you can find the ratio is

n

N

=NA= 6. 0 × 1023 [ 8 ]

.

Question

Assuming the atmosphere is isothermal (at constant temperature), what will happen to a perfectly elastic (no

elastic force,unlikethe example above) gas balloon as it floats higher and higher?

Answer

A perfectly elastic balloon is in equilibrium when the gas pressure inside matches the air pressure outside. As

T=const, we can use Boyle’s law, that is,PV=const. As the balloon rises higher, the air pressure outside it

drops, and therefore the gas pressure inside must drop as well. By Boyle’s law, we determine that its volume

must increase.

Question

Explain why this result also holds for a balloon that is not perfectly elastic.

Diagrams of Quasistatic Gas Processes

Using the formulas above, it is possible to graphically display processes suggested by the laws above. For instance,

consider the diagram on the left below, which shows an ideal gas being heated and compressed onT−Vaxes. As

we will see later, it is often useful to graph such processes on a different set of axes, sayP−V, as shown on the

right.