GASES

Gases The physical state of gases is defined by several physical properties Volume Temperature Amount (commonly expressed as number of moles) Pressure p = f(T, V, n)

Equation of state An equation of state of a gas relates the four variables describing the physical properties of the gas p = f(T, V, n)

THE Perfect Gas

The perfect gas A gas that obeys the perfect gas equation The product of pressure and volume is proportional to the product of amount and temperature PV = nRT

For a fixed amount of gas (constant n) plot of the properties of a gas gives a surface Isobar - pressure constant - line, V a T Isotherm - temperature constant, hyperbola, PV = constant Isochore- volume constant - line P a T

Kinetic Model of Gases

𝑐 = root mean square speed of the gas molecules 𝑝𝑉= 1 3 𝑛𝑀 𝑐 2 𝑐 = root mean square speed of the gas molecules 𝑐= 3𝑅𝑇 𝑀 The root mean square speed of the molecules of a gas is proportional to the square root of the temperature and inversely proportional to the square root of the molar mass

Maxwell distribution of speeds

Exercises What is the pressure if 1 mole of N2 occupies 1 L of volume at 1000 K? At standard temperature and pressure, how many grams of CO2 is contained in a 3.0 L container?  What is the density of acetone, C3H6O, vapor at 1.0 atm and 400 K? The density of acetone is 1.767 g/L, calculate its molar mass.

Real Gases

Compressibility factor z Also known as the compression factor Employed as a measure of the ideality of gases z ≅ 1 if perfect gas At very low pressures, real gases behave like a perfect gas and z ≅ 1 𝑃𝑉=𝑧𝑛𝑅𝑇 𝑧= 𝑃𝑉 𝑛𝑅𝑇

Compressibility factor z It is expressed as the ratio of the measured molar volume and the molar volume of a perfect gas 𝑧= 𝑉𝑚 𝑉°𝑚 Where 𝑉𝑚=𝑉/𝑛 Since in an ideal gas, 𝑉°𝑚= 𝑅𝑇 𝑝 Therefore 𝑧= 𝑉/𝑛 𝑅𝑇/𝑝 And 𝑧= 𝑝𝑉 𝑛𝑅𝑇

z > 1 Happens at high pressures Repulsive forces between gas molecules are now dominant As a result, the real gas has a larger molar volume than the corresponding ideal gas

z < 1 Happens at low pressures Attractive forces between gas molecules are now dominant As a result, the molar volume of the real gas is now smaller than that of the corresponding ideal gas

Selected equations of state * Please take note that the equations here assume 1 mole of the substance, hence the use of Vm. For proper calculations, use the standard and default equations.

Surface of possible states for a gas obeying the Van der Waal’s equation Surface of possible states for an ideal gas

Virial equations and coefficients The virial equation shows that as pressures approach zero (p→0), it coincides with the ideal gas law. At low pressures, usually the first term of the virial equation is necessary. At higher pressures, the other terms become more significant.

Boyle Temperature The temperature wherein the properties of a real gas coincide with those of a perfect gas as p→0 Can also be defined as the temperature where a real gas obeys the ideal gas law over an appreciable pressure range. Deviations of z from unity are positive when the gas is above the Boyle temperature Below the Boyle temperature, the value of z decreases below unity before approaching it again and increasing above 1.

Critical constants A gas can be condensed through compression when its below the critical temperature. If not, it will just form a supercritical fluid. These are found at the critical point of a gas: Critical temperature (TC) Critical pressure (PC) Critical volume (VC)

Principle of corresponding states We can use a certain property and set up a relative scale for the purpose of comparing the properties of objects The critical constants are characteristic properties of gases and we can use them to set up a scale. This leads to the reduced variables, which can be found by dividing the actual variable with the corresponding critical constant.

Principle of corresponding states

Principle of corresponding states The principle of corresponding states postulates that real gases at the same reduced volume and reduced temperature exert the same reduced pressure. It does not follow that this is true for all conditions as the principle is only an approximation, but it does suggest that at their corresponding states, frequently better correlation of experimental data may be obtained.

Problems The density of dry air at 750 torr and 28°C is 1.145 g/L. What is the composition of dry air, assuming that only nitrogen gas and oxygen gas are present? A 19.5 L flask at 15 °C contains a mixture of three gases: N2 (2.50 mol), He (0.38 mol), and Ne (1.34 mol). Calculate the partial pressure of neon gas in the mixture.