Gas Mixtures Partial Pressures and Mole Fractions

Click Coached Problems for a self-study module on partial pressures. 

Because the ideal gas law applies to all gases, you might expert it to apply to gas mixtures. Indeed it does. For a mixture of two gases A and B, the total pressure is given by the expression 

Partial pressure is the pressure a gas would exert if it occupied the entire volume by itself. 

The terms nARTIV and nBRTIV are, according to the ideal gas law, the pressures that gases A and B would exert if they were alone. These quantities are referred to as partial pressures, PAandPu. 

PA = partial pressure A = nA RTIV PB = partial pressure B = nB RTIV 

The relation just derived was first proposed by John Dalton in 1801 it is often referred to as Dalton s law of partial pressures  

The total pressure of a gas mixture is the sum of the partial pressures of the components of 

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The amount of gas introduced into a diving tank can be determined by weighing the tank before and after charging the tank with gas. A diving shop placed 80.0 g of O2 and 20.0 g of He in a 5.00-L tank at 298 K. Determine the total pressure of the mixture, and find the partial pressures and mole fractions of the two... 

C05-0025. Find the partial pressures and mole fractions of a gas mixture that contains 1.00 g each of H2 and N2, if the total pressure of this mixture is 2.30 atm. 

Calculating Partial Pressures and Mole Fractions of a Gas in a Mixture... 

Calculating partial pressures and mole fractions of a gas in a mixture Given the masses of gases in a mixture, calculate the partial pressures and mole fractions. (EXAMPLE 5.10)... 

The ideal gas law can be rearranged to calculate the density and molar mass of a gas. In a mixture of gases, each component contributes its own partial pressure to the total pressure (Dalton s law of partial pressures). The mole fraction of each component is the ratio of its partial pressure to the total pressure. When a gas is in contact with water, the total pressure is the sum of the gas pressure and the vapor pressure of water at the given temperature. 

Other composition variables are sometimes used, such as volume fraction, mole ratio, and mole percent. To describe the composition of a gas mixture, partial pressures can be used (Sec. 9.3.1). 

Apply the ideal gas law to determine the density of a gas at different temperatures, the molar mass of a gas, and the partial pressure (or mole fraction) of each gas in a mixture... 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

Raoul s equation states that the partial pressure of a component in the gas is equal to the mole fraction of that component in the liquid multiplied by the vapor pressure of the pure component. Raoult s equation is valid only if both the gas and liquid mixtures are ideal solutions. The mathematical statement is... 

Sj. Mixtures of Real Gases Additive Pressure Law.—The rule that the total pressure of a mixture of gases is equal to the sum of the pressures exerted by each gas if it alone occupied the whole of the available volume ( 5b) does not apply to real gases. The total pressure is thus not equal to the sum of the partial pressures defined in the usual manner. However, for some purposes it is convenient to define the partial pressure of a gas in a mixture by means of equation (5.8), i.e., p == n P, where p is the partial pressure and n is the mole fraction of any constituent of the mixture of gases of total pressure P. 

Dalton s law for ideal gas mixtures is used to express partial pressures as a product of total pressure in atmospheres and mole fraction y,. Based on the definition of CO conversion x in terms of molar flow rate for gas-phase flow reactors and the fact that the mole fraction of component i is equal to its molar flow rate divided by the total molar flow rate, the following expression is obtained for the mole fraction of component i ... 

Here Xi is the mole fraction of i in the liquid mixture, and pf and f are the partial pressure and fugacity in a gas phase equilibrated with pure liquid i at the same T and p as the liquid mixture. Both p" and f are functions of T and p. 

Practice Problem B Determine the partial pressure and number of moles of each gas in a 15.75-L vessel at 30.0°C containing a mixture of xenon and neon gases only. The total pressure in the vessel is 6.50 atm, and the mole fraction of xenon is 0.761. 

The design becomes somewhat more complex when a mixture of compounds is involved. However, the components of the mixture are usually mutually soluble in the liquid phase, and, as a first approximation for related solvents, it can be assumed that the mixture follows Raoult s law, i.e., the partial pressure of each component in the product gas will be equal to the vapor pressure of the pure component at the gas outlet temperature times its mole fraction in the liquid phase. For more precise calculations and more complex liquid mixtures, it is necessary to use vapor-liquid equilibrium (VLB) data for the specific system. The estimation and correlation of VLB data are discussed in various chemical engineering texts, such as Perry s Handbook (Perry et al., 1984), Reid and Sherwood (1966), and Prausnitz (1969). 

The equihbrium partitioning of a chemical solute between a Hquid and vapor phase is governed by Henry s law when the Hquid mixture is very dilute in the solute. Henry s law generally is vaHd at concentrations below 0.01 mol/L of solution, although the upper limit can sometimes extend to 0.1 mol/L or higher (10). Over this concentration range, a direct proportionaHty, ie, Henry s constant, is observed between the partial pressure of the chemical in the gas phase and its mole fraction in the Hquid phase. Henry s constant, when expressed in this way, has units of pressure (3). 

Assuming an ideal gas mixture at amiospheric pressure, calculate llie mole fraction and ppm of a component if its partial pressure is 19 mniHg. 

Thus, in an ideal gas mixture, the mole fraction of each component is identical with its volume fraction (by Amagat s law) or the ratio of its partial pressure to the total pressure (by Dalton s law). For both laws to be applicable simultaneously, the mixture and its components must behave ideally. 

In other words, the partial pressure of a gas in a mixture is equal to its mole fraction multiplied by die total pressure. This relation is commonly used to calculate partial pressures of gases in a mixture when the total pressure and the composition of the mixture are known (Example 5.9). 

The easiest way to express the relation between the total pressure of a mixture and the partial pressures of its components is to introduce the mole fraction, x, of each component A, B,. . ., the number of moles of molecules of the gas expressed as a fraction of the total number of moles of molecules in the sample. If the amounts of gas molecules present are nA, B, and so forth, the mole fraction of A is... 

To find the relation between the partial pressure of a gas in a mixture and its mole fraction, we first express the partial pressure, Pa, of a gas A in terms of the amount of A molecules present, wA, the volume, V, occupied by the mixture, and the temperature, T ... 

STRATEGY To use Eq. 14, we need the total pressure (given) and the mole fraction of each component. The first step is to calculate the amount (in moles) of each gas present and the total amount (in moles). Then calculate the mole fractions from Eq. 12. To obtain the partial pressures of the gases, multiply the total pressure by the mole fractions of the gases in the mixture (Eq. 14). 

A sample of hydrogen chloride gas, HC1, is being collected by bubbling it through liquid benzene into a graduated cylinder. Assume that the molecules pictured as spheres show a representative sample of the mixture of HC1 and benzene vapor ( represents an HCl molecule and O a benzene molecule), (a) Use the figure to determine the mole fractions of HCl and benzene vapor in the gas inside the container, (b) What are the partial pressures of HCl and benzene in the container when the total pressure inside the container is 0.80 atm ... 

Each gas establishes its own dynamic equilibrium with water. The concentration depends on the partial pressure of the gas in the atmosphere and on the value of its Henry s law constant at 25 °C. Recall from Chapter 5 that the partial pressure of any gas in a mixture is given by the mole fraction (X multiplied by total pressure. 

If you know how many moles of each gas are in the mixture and the total pressure, you can calculate the partial pressure of each gas by multiplying the total pressure by the mole fraction of each gas ... 

The description of the partial pressure exerted by a sorbate, or a mixture of sorbates, when they reside on the sorbent surface, at some given temperature is what we speak of as adsorption equihbrium. For a single adsorbate (adsorbing molecular species) we require three state variables to completely describe the equilibrium the temperature, the sorbed phase concentration or loading and the partial pressure exerted by the sorbed phase are very convenient variables to use. As more adsorbable compounds are added to the problem we require additional information to adequately describe the problem. That information is the specification of the mole fractions of the adsorbable compounds in both the gas and sorbed states. 

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