Gas Laws

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 = Vo+ 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 (Vt is the volume of the gas at some temperature, Vo is the volume at 0 oC):

Vt/Vo = 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

tV=0 = - 1/c ,

or, from the preceding equation:

tV=0 = - Vo/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 oC, and

T(K) = t(oC) + 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.

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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 m3-Pa/mol-K).

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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'P2 + ...]  .

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 + an2/V2)(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.

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In this simulated experiment, a temperature between 0 and 500 oC 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.

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