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Gas Mixtures
Mixtures of gases can be described in terms of volume, pressure or mole fractions.

The ideal-gas expressions PV = nRT can be applied to gases that are pure, single-component gases or are mixtures of different components. For gas mixtures it is sometimes necessary to relate the properties of the gas mixture to those of its components.

Partial pressures: the pressure needed to confine the individual components of a gas sample separately, and together, in a chamber of fixed volume can be determined. If the pressure P needed to confine the gas mixture is found to be equal to the sum of the individual pressures P1, P2 ….. needed to confine each component by itself, the gas mixture is said to obey Dalton’s law of partial pressures. We can write this law as

P = P1 + P2 + P3 + …… = ΣPi

Dalton’s law of partial pressures is followed closely by many mixtures of gases. It is, however, another “ideal-gas” law. It is approximately obeyed by real-gas mixtures under ordinary conditions. But all real-gases mixtures behave in accordance with this law as the total pressure on the gas mixture approaches zero.

Dalton’s law is expected if each component of the gas mixture and the mixture itself follow the ideal-gas laws summarized by PV = nRT. Let n1, n2, n3 ….. be the number of moles of the number of moles of the various components, and let n be the total number of moles of molecules in the gas mixture. Then we can write

n = n1 + n2 + n3 + …….

The ideal-gas expression can be written for the mixture and for each component. We have

n = PV/RT  n1 = P1V/RT  n2 = P2V/RT  ……

substitution for the mole terms in the above equation gives

PV/RT =  P1V/RT  + P2V/RT + ……

Or P = P1 + P2 + ……


Ideal behavior of the gas components and the gas mixture implies obedience to Dalton’s law of partial pressures.

Dalton’s law is followed quite closely by many mixtures of gases. You should note, however, that even if the separate gases behave ideally, the mixture might not, and it might not confirm to Dalton’s law. Obvious examples are provided by gas mixtures in which ideal-gas components react with one another.

Partial volumes: it is sometimes convenient to think of a gas mixture in terms of the volumes that the components would occupy if they were pure gases at the total pressure of the gas mixture.

We can start again with the description of the gas mixture given by m = n1 + n2 + ….. but now the gas mixture and the components are thought of as being confined by the total pressure P. the equation PV = nRT, applied to the gas mixture and to each component, then yields

V = V1 + V2 + ……

This expression suggests that the volume of a gas mixture is equal to the sum of the volumes of the components, with each component and the gas mixture being at the total pressure P. this result is applicable to the same as Dalton’s law.

It is convenient sometimes to think of gas mixtures in terms of the partial pressure of the components and at other times to think in terms of the partial volumes of the components.

Mole fractions: in dealing with the mixtures we frequently use the fractional contributions of each component to some total property of the system. Such contributions can be expressed in terms of the mole fraction xi of the ith component. If ni is the number of moles of the molecules of the ith component in a particular sample and n is the total number of moles of molecules, the mole fraction of the ith component is defined by

xi = ni/n    

Writing n = n1 + n2 + n3 + …. And then dividing both sides by n, we obtain

1 = n1/n + n2/n +  n3/n + …..

Or 1 = x1 + x2 + x3 + …..

The sum of fractional quantities, like the mole fraction, over all the components of the mixture is unity.

The pressure of fraction Pi/P and the volume fraction Vi/V can be dealt with in a similar way.

When the gas mixture and the components are ideal and Dalton’s law is followed, the mole fraction, the pressure fraction, and the volume fraction are all equal. You can see this by writing

Pi/P  = (ni RT/V)/(n RT/V) = ni/n = xi   

And Vi/V = (ni RT/P)/(n RT/P) = ni/n = x  

Mass of 1 mol of molecules of a gas mixture: the mass of 1 mol of gas molecules can be determined from a measured value of the gas density.

M = x1M1 + x2M2 + x3M3 + ……

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