**In the mid-1600's, Robert Boyle conducted experiments by trapping
air above mercury in a sealed J-tube, measuring the volume as a function
of pressure. He concluded (Boyle's Law)
that for each trapped sample, the product of pressure x volume was constant:**

**PV = constant .**

**[Each experiment was conducted over a short period of time,**
**so that changes of temperature were very small.]**

**About 100 years later, Jacques Charles measured the effect of temperature
on the volumes of gases at constant pressure. His results were published
17 years later by Joseph Louis Gay-Lussac. Two very important relationships
were established by this work:**

1. At constant pressure, the volume of a gas changes linearly with temperature.

V = V_{o}+ k t .

2. The same volumes of different gases at the same temperature change by the same amount when taken to another temperature. This led to the relationship (V_{t}is the volume of the gas at some temperature, V_{o}is the volume at 0^{o}C):

V_{t}/V_{o}= 1 + c t ,

in which c had the same value for different gases.

**Gay-Lussac noted that the volume would become zero when the temperature
reached**

**t _{V=0} = - 1/c ,**

**or, from the preceding equation:**

**t _{V=0} = - V_{o}/k
.**

**William Thomson (Lord Kelvin) later used these results to propose
an absolute-temperature scale (the Kelvin scale),
in which absolute zero equals -273.15 ^{o}C, and**

**T(K) = t( ^{o}C) + 273.15 .**

**Using this scale of temperature, the volume of a trapped sample of
gas at constant pressure is directly proportional to temperature:**

**V = constant x T .**

**This relationship is now called Charles' Law
or the Law of Charles and Gay-Lussac.**

**Gay-Lussac also conducted extensive research on the relationships
between volumes and masses of gases, and on the changes in volume as gases
reacted. Amadeo Avogadro used those observations to formulate his
hypothesis:**

**Equal volumes of different gases at the same
temperature and pressure**
**contain equal numbers of molecules.**

**This formed the basis for our present understanding of moles and
molar masses.**

**_________________**

**The works of Boyle, Charles, Gay-Lussac, Avogadro, and Kelvin have
been combined into the IDEAL GAS LAW:**

**PV = nRT ,**

**in which R is a universal consatnt (R = 0.08206 L-atm/mol-K or R
= 8.314 m ^{3}-Pa/mol-K).**

**_________________**

**While the ideal gas law is an extremely useful approximation for
calculations on the properties of gases, and is exact for gases at low
pressures, careful measurements show that most gases deviate progressively
from this relationship at higher pressures. The actual behavior is
often described with the virial equation:**

**PV = nRT[1 + B(n/V) + C(n/V) ^{2} +
...] ,**

**in which the temperature-dependent constants for each gas are known
as the virial coefficients. The second virial coefficient, B , has
units of molar volume (L/mole).**

**This is sometimes replaced with a modified version with different
coefficients,**

** PV = nRT[1 + B'P + C'P ^{2} +
...] .**

**The values for B(L/mole) and B'(atm ^{-1})
are mathematically related,**

** B = RT B' .**

**Other equations have been developed to describe real gases, particularly
with respect to their liquid - vapor properties, such as the van
der Waals equation (each gas has specific values for the constants
a and b):**

** (P + an ^{2}/V^{2})(V
- nb) = nRT ;**

**This equation relates the second-virial coefficient (B) of a gas
to the van der Waals constants (a,b):**

**B = b - a/RT
,**

**and predicts that the second virial coefficient of a gas should be
negative at low temperatures, becoming less negative and possibly positve
with increasing temperature. There is a specific temperature for
each gas where its second virial coefficient will be zero, and the gas
will obey Boyle's Law to rather high pressures. This temperature
is called the Boyle temperature for the gas.**

**_________________**

**In this simulated experiment, a temperature between 0 and 500 ^{o}C
may be chosen and there is a menu of five gases. When a gas is selected,
the cylinder is emptied then re-filled to a randomly-selected volume at
the selected temperature and a pressure of 1.000 atm. The number
of moles of gas is not known. By measuring the pressure-volume relationship
at various pressures at constant temperature, the data may be fitted to
an straight line and the slope and intercept may be determined:**

**PV = intercept + slope x P .**

**Comparison to the equation above (dropping the higher-order terms):**

** PV = nRT[1 + B'P]
,**

**shows that the intercept is**

** intercept =
nRT ,**

**and the slope is**

** slope = nRTB'
.**

**The value of B' can be calculated by
dividing the slope by the intercept:**

**B' = slope/intercept
.**

**The virial coefficient B may be calculated
by multiplying B' by RT,
and the number of moles in the sample (n)
may be calculated by dividing the intercept
by RT.**