Gases and Vapor
Ideal gas Law
The ideal gas law is derived from the kinetic theory of gases. This law relates the molar quantity and volume of a gas to temperature and pressure. The simplest form of these relationships is as follows:
where
P = absolute pressure of a gas (N/m²)
V = volume of the gas (m³)
n = number of moles of the gas (mol)
= Universal gas constant (N·m/mol·K, values are depending on the units of P, V, n, and T.)
T = absolute temperature of the gas (K)
It is adequate for many engineering calculations involving gases at low pressure. Under conditions of high pressure and temperature all gases deviate from this ideal behavior. In such conditions, PVT calculations use more complex equations.
The gas constant per unit mass, R, is defined as follows;
or
where
m = mass of the system (kg)
R = gas constant per unit mass (N·m/kg·K)
Recalling that m/n is equal to molecular weight of the gas, then;
Substituting into the ideal gas equation gives,
PV=mRT
Table 1. The Universal Gas Constant Expressed in Various Units
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= 1545 ft lbf/lbm-mole R = 0.082055 liter atm/mol K
= 1.986 Btu/lbm-mole R = 8.314 N m/mol K
= 1.986 cal/mol K = 8.314 x 103 N m/kmol K
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Dalton’s Law
Dalton’s law states that if a gas mixture is contained in a given volume V at a temperature T, the total pressure of the mixture is equal to the sum of the partial pressure of each gas species.
Vapor Pressure:
Vapor pressure is the pressure when T and P correspond to a point on the vapor-liquid equilibrium curve for a substance while T is said to be the boiling temperature. "The vapor pressure of a species is by measure of its volatility"(Felder, 2000). The volatility is the degree that the species tends to transfer from the liquid state to vapor state. Higher the vapor pressure at a given temperature, greater the volatility of the species at that given temperature.
The pressure - temperature diagram for water is shown in Figure 1. Water exists in ice, liquid, and vapor phases depending on the pressure and temperature. Lines separate the phases in Figure 1. Along the lines, water can exist at the two different phases. Thus considering line A-C for any one temperature along the line there is only one pressure at which water is in equilibrium. Points along line A-C are called saturated. Point C is the critical point beyond which the liquid and vapor phases are not distinguishable.
Figure 1. Pressure - Temperature diagram for water.
One unique feature of the solid - liquid curve A-D is that an increase in pressure lowers the melting point. A practical example is the pressure under the blades of ice skates provides a liquid layer of water for lubricating the blade. Water can simultaneously exist as a solid, liquid and vapor at the triple point, point A.
Curve A-B is the sublimation-pressure curve and separates the ice and vapor phase regions.
The thermodynamic surface for a pure substance is three-dimensional with coordinates of pressure, temperature, and volume i.e. P-?-T. The volume v coordinate is perpendicular to the P-T view shown in Figure 1 above. The P-? diagram for line A-C is shown in Figure 2. Pressure, kPa 
Figure 2. Pressure - volume diagram for water.
Water along line A-C may exists in equilibrium as a liquid and vapor mixture. The liquid has the properties of a saturated liquid and the vapor has the properties of a saturate vapor. The thermodynamic properties of the mixture are proportional to the respective mass fractions of the mixture. The total mass (m) of a mixture is sum of the vapor (mg) and the liquid fractions (mf)
m = mf + mg
and the total volume is the sum of the liquid and vapor volumes
V = Vf + Vg
And by using the definition of specific volume
? = m/V
the total volume can be expressed as
m ? = mf ?f + mg ?g
Dividing by m gives
? = (m
f /m) ?f + (mg /m)/ ?gThe mass ratio vapor to total mass is defined as the quality x of the mixture. Thus
x = mg/m
Likewise
(1 - x) = mf/m
These relationships for quality can be substituted into the above equation to give
? = (1 - x) ?
f + x ?gwhich rearranges to
? = ?
f + x (?g - ?f)
The difference between the vapor and liquid volumes are conveniently tabulated in many steam tables and designated as ?fg
?fg = (?g - ?f)
Substituting into the above gives the following useful relationship:
? = ?
f + x ?fg
The mixture properties of enthalpy, entropy, and internal energy are similarly determined.
Steam tables
The thermodynamic properties of enthalpy, specific volume, internal energy, and entropy are tabulated as a function of pressure or temperature in steam tables. Saturated steam tables give thermodynamic properties for points along the line A-C. Superheated steam tables are tabulated as a function of both pressure and temperature because they are independent properties in this region. The saturated solid-vapor table gives the properties along line A-B of Figure 1.
Raoult's Law
Pure water has a vapor pressure that varies with temperature as shown in Figure 1. Pure liquids, if considered an ideal solution, will have uniform cohesive forces between molecules. Solutes act to lower the vapor pressure in liquids. Raoult's Law describes the relationship of vapor pressure of a liquid with solute concentration as follows:
where
This equation applies only to non-disassociating chemical species. Salt (NaCl) disassociates into Na+ and Cl- in water and thus the number of moles in the solute doubles. Likewise, the number of moles in solution triples for CaCl2. The equilibrium relative humidity above a water solution is equivalent to the mole fraction of water and is calculated as follows:
where
Vapor pressure of water above a solution
at a temperature T.
Vapor pressure of pure water at a
temperature T.
Example 1.
What is the equilibrium relative humidity above a solution containing 60% sucrose (MW = 342) and 50% water by weight?
Solution:
Assume a basis of 1 g of solution. The moles of sucrose and water are calculated as follows
The equilibrium relative humidity in a confined space above this solution is thus 92.6%.
If the temperature is 25ºC then the vapor pressure of pure water at saturation is found in a steam table to be 3.169 kPa. The vapor pressure of water, P, above the solution is thus
It can be seen from Raoult's Law that low molecular weight solutes are more effective than higher molecular weight solutes in reducing the vapor pressure per unit mass. Toledo (1991) discusses the non-ideal nature of water in foods and its effect of the equilibrium relative humidity of water.
