Page 34 - Physical Chemistry

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What happens if we vary the mass m of ideal gas while keeping P and T constant? Section 1.5
Volume is an extensive quantity, so V is directly proportional to m for any one-phase, Ideal Gases
one-component system at constant T and P. Thus V/m is constant at constant T and P.
Combining this fact with the constancy of PV/T at constant m,we readily find (Prob.
1.24) that PV/mT remains constant for any variation in P, V, T, and m of any pure ideal
gas: PV/mT c, where c is a constant. There is no reason for c to be the same for dif-
ferent ideal gases, and in fact it is not. To obtain a form of the ideal-gas law that has
the same constant for every ideal gas, we need another experimental observation.
In 1808 Gay-Lussac noted that the ratios of volumes of gases that react with one
another involve small whole numbers when these volumes are measured at the same
temperature and pressure. For example, one finds that two liters of hydrogen gas react
with one liter of oxygen gas to form water. This reaction is 2H O → 2H O, so the
2
2
2
number of hydrogen molecules reacting is twice the number of oxygen molecules re-
acting. The two liters of hydrogen must then contain twice the number of molecules
as does the one liter of oxygen, and therefore one liter of hydrogen will have the same
number of molecules as one liter of oxygen at the same temperature and pressure. The
same result is obtained for other gas-phase reactions. We conclude that equal volumes
of different gases at the same temperature and pressure contain equal numbers of mol-
ecules. This idea was first recognized by Avogadro in 1811. (Gay-Lussac’s law of
combining volumes and Avogadro’s hypothesis are strictly true for real gases only in
the limit P → 0.) Since the number of molecules is proportional to the number of
moles, Avogadro’s hypothesis states that equal volumes of different gases at the same
T and P have equal numbers of moles.
Since the mass of a pure gas is proportional to the number of moles, the ideal-gas
law PV/mT c can be rewritten as PV/nT R or n PV/RT, where n is the number
of moles of gas and R is some other constant. Avogadro’s hypothesis says that, if P,
V, and T are the same for two different gases, then n must be the same. But this can
hold true only if R has the same value for every gas. R is therefore a universal con-
stant, called the gas constant. The final form of the ideal-gas law is
PV nRT ideal gas (1.18)*

Equation (1.18) incorporates Boyle’s law, Charles’ law (more accurately, the defini-
tion of T), and Avogadro’s hypothesis.
An ideal gas is a gas that obeys PV nRT. Real gases obey this law only in the
limit of zero density, where intermolecular forces are negligible.
Using M m/n [Eq. (1.4)] to introduce the molar mass M of the gas, we can write
the ideal-gas law as
PV mRT>M ideal gas

This form enables us to find the molecular weight of a gas by measuring the volume
occupied by a known mass at a known T and P. For accurate results, one does a series
of measurements at different pressures and extrapolates the results to zero pressure
(see Prob. 1.21). We can also write the ideal-gas law in terms of the density r m/V as

P rRT>M ideal gas
The only form worth remembering is PV nRT, since all other forms are easily
derived from this one.
The gas constant R can be evaluated by taking a known number of moles of some
gas held at a known temperature and carrying out a series of pressure–volume mea-
surements at successively lower pressures. Evaluation of the zero-pressure limit of
PV/nT then gives R (Prob. 1.20). The experimental result is
3
R 82.06 1cm atm2>1mol K2 (1.19)*