Properties of Gases Chapter 1

1. Properties of GasesChapter 1 CHEM 321-01 Thermodynamics and Equilibrium Daniel E. Autrey, Ph.D.

2. Phases of Matter There are three basic phases of matter: • Gas • Particles are separated by distances that are large compared with the size of the molecules. Gases completely fill the container they occupy, taking on the shape of the container. Intermolecular interactions are minimal. • Liquid • Particles are close together but are not held rigidly in position and can move past one another. Liquids do not completely fill the container they occupy but do take on the shape of the container. • Solid • Particles are held close together in an orderly fashion with little freedom of motion. Solids do not completely fill a container and do not take on the shape of the container they occupy.

3. Ideal Gases Many useful thermodynamic principles are illustrated by considering ideal, or perfect, gases. • Ideal Gas • A hypothetical gas, or collection of molecules or atoms, which undergo continuous random motion (or Brownian motion). • Characteristics: • The speeds of the particles increase with increasing temperature. • The molecules are widely separated from each other, with the only interactions being infrequent elastic collisions with the walls of the container and other particles. • The particles do not experience any intermolecular forces, such as dipole-dipole forces or dispersion forces. • Particles are considered to be “point-masses,” having mass, but no volume.

4. States of Gases • Physical State – The physical condition of a sample of a substance that is defined by its physical properties. • The state of a pure gas can be specified by giving its: • Volume (V) • Amount of substance in moles (n) • Pressure (P) • Temperature (T) • Equation of State – An equation that relates these four variables, such as: • Experimentally, if we know any three of these variables, then we can solve for the fourth.

5. Pressure • Pressure (p) – The force applied per unit area. • Pressure exerted by a gas is caused by the collisions of the gas molecules with the walls of the container. The more collisions, the greater the pressure. • The SI unit of pressure is the pascal (Pa). • 1 Pa = 1 N m-2 • 1 Pa = 1 kg m-1 s-2 • Standard pressure (po) is a pressure of 105 Pa (or 1 bar). • Other common units of pressure: • Atmosphere (atm) 1 atm = 101,325 Pa • Torr (Torr) 760 Torr = 1 atm 1 Torr = 133.32 Pa • Millimeter of mercury (mmHg) 1 mmHg = 1 Torr 1 mmHg = 133.32 Pa • Pound per square inch (psi) 1 psi = 6.894757 kPa

6. Mechanical Equilibrium • Consider two gases at different pressures that are separated by a movable wall (or piston). • The higher-pressure gas will move the wall and compress the lower- pressure gas, until an equilibrium pressure is established. • The condition of equality of pressure on either side of the moveable wall is a state of mechanical equilibrium between the two gases.

7. Measuring Pressure • Atmospheric pressure is measured using a device known as a barometer,invented by Torricelli, which is an inverted tube of mercury sealed at the upper end. When the column of mercury is in mechanical equilibrium with the atmosphere, the pressure at its base is equal to that exerted by the atmosphere. The height of the mercury column is proportional to the external pressure.

8. Measuring Pressure • A manometer is a device for measuring pressures of samples of gases, in which a nonvolatile viscous fluid is contained in a U-tube, with one end open to the atmosphere.

9. Temperature • Temperature (T) – property that indicates the flow of energy (in the form of heat) through a thermally conductive, rigid wall. Energy flows from higher temperatures to lower temperatures. • Types of Boundaries: • Diathermic – Allows the flow of energy between two contacting objects resulting in a change of state to occur. • Adiabatic – Does not allow the flow of energy between two contacting objects.

10. Thermal Equilibrium • Thermal energy flows from an object of high temperature to an object of low temperature if the objects are in contact through a diathermic wall. • The flow of thermal energy will continue until both objects are of equal temperatures and reach a state of thermal equilibrium. • Thermal equilibrium is established if no change of state occurs when two objects are in contact with one another through a diathermic boundary.

11. Zeroth Law of Thermodynamics • Zeroth Law of Thermodynamics - If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C is also in thermal equilibrium with A. • Allows for the use of a thermometer to measure temperature.

12. Boyle’s Law • Boyle’s Law – the volume of a gas is inversely proportional to the pressure at constant temperature. (Robert Boyle, 1661) • An isotherm is a plot of pressure versus volume at constant temperature.

13. Boyle’s Law • A plot of pressure versus 1/V results in a straight line. • Boyle’s Law is a limiting law, in that it is valid only in the limit of low pressures for real gases. • The reason that Boyle’s Law applies to all gases at low pressure regardless of their chemical identity is because at low pressure the average separation of molecules is so great that they exert no influence on one another and travel independently.

14. Gay-Lussac’s Law • Gay-Lussac Law – the volume of a gas increases linearly with temperature when the pressure is kept constant. (Joseph Louis Gay-Lussac, 1802) • May be expressed as: • T is the temperature in C. • V0is the volume of gas at 0C. •  is the proportionality constant, also known as the coefficient of thermal expansion. • Isobar – variation of volume and temperature at constant pressure.

15. Gay-Lussac’s Law • Using the Kelvin temperature scale, the Gay-Lussac Law is written as: • Alternatively, at constant volume, the Gay-Lussac’s Law may be expressed as: • Why? • As the temperature of the gas increases, the average molecular speed increases, resulting in molecules colliding with the walls of the container with greater impact, thus increasing the pressure.

16. Avogadro’s Principle • Avogadro’s Principle – equal volumes of gases at the same pressure and temperature contain the same number of molecules. • This is based on the observation that at a given pressure and temperature, the molar volume , the volume per mole of molecules, is approximately the same regardless of the identity of the gas. • At constant pressure and temperature:

17. Ideal Gas Law • Combining Boyle’s Law, Gay-Lussac’s Law, and the Avogadro’s Principle gives the proportionality: • This may also be written as the ideal gas equation: • R is the constant of proportionality, and is called the gas constant. Experimentally, it is found to be the same for all gases. • R has different values depending on the units used:

18. Ideal Gas Law • The ideal gas equation represents the approximate equation of state for any gas, and becomes increasingly exact as the pressure approaches zero. • A real gas, or actual gas, behaves like an ideal gas at the limit of zero pressure. • The ideal gas equation is very useful in calculating the properties of gases under different conditions. • For example, the molar volume, Vm, of a gas is calculated to be: • 24.789 L mol-1 at Standard Ambient Temperature and Pressure (SATP), which is 298.15 K and 1 bar. • 22.414 L mol-1 at Standard Temperature and Pressure (STP), which is 0 C and 1 atm.

19. Ideal Gas Law • For a fixed amount (constant n) of a gas, we can plot a surface as a function of pressure, volume, and temperature. • The surface represents the only states of the gas that can exist.

20. Combined Gas Law • The ideal gas equation can be used to calculate the change in conditions when a fixed amount of gas is subjected to different temperatures and pressures and allowed to occupy a different volume. • Since under one set of conditions: • And under another set of conditions: • It follows that: which is known as the combined gas law.

21. Gaseous Mixtures • To understand the physical behavior of a gaseous mixture, we need to know the contribution that each component makes to the total pressure of the sample. • Dalton’s Law of Partial Pressures – the pressure exerted by a mixture of gases is the sum of the partial pressures of the gaseous components. • Mathematically: where pAand pB are the partial pressures of the perfect gases A and B, and pis the total pressure of the mixture. • The partial pressure of a perfect gas is the pressure it would exert if it occupied the container alone at the same temperature, determined from the ideal gas equation:

22. Mole Fraction • Mole Fraction – the amount of molecules of a component (J) of a mixture expressed as a fraction of the total amount of molecules (n) in the sample: where • The sum of the mole fractions of the components of a mixture is unity, or: • The partial pressure of any gas, not necessarily an ideal gas, is related to the mole fraction by:

23. Mole Fraction • Dalton’s Law may be expressed in terms of the mole fraction of the components and is valid for any gas:

24. Real Gases • Real Gas • An actual gas that does not obey the ideal gas law exactly because of molecular interactions. The deviations occur at high pressure and low temperatures. • Two types of interactions: • Repulsive forces – • Short-range interactions • Occurs at high pressure when the molecules are nearly in contact with each other. • Makes the gas less compressible than an ideal gas because the repulsive forces help to drive the molecules apart. • Attractive forces – • Long-range interactions • Occurs at lower temperature when the molecules move so slowly that they can be captured by one another. • Makes the gas more compressible than an ideal gas because the attractive forces tend to pull the molecules together.

25. Compressibility • Compressibility • A measure of the nonideality of real gases due to the presence of either repulsive or attractive forces. • Expressed by the compression factor, Z, which is the ratio of its molar volume to the molar volume of an ideal gas at the same pressure and temperature: • Vm is the molar volume of the real gas • is the molar volume if the gas were ideal • From the ideal gas equation: • Therefore,

26. Compression Factor • If Z = 1, • The gas is ideal. • No intermolecular forces. • If Z > 1, • Occurs at high pressure. • Repulsive forces dominate, causing expansion. • If Z < 1, • Occurs at intermediate pressure. • Attractive forces dominate, causing compression.

27. Virial Equation • Virial Equation - An equation of state for real gases that expresses the compression factor as a power series of either the pressure or molar volume. • “Virial” comes from the Latin word, viris, meaning force. • Experimentally, it has been found that the compression factor varies almost linearly with pressure: • B' is a temperature-dependent constant for a specific gas. • As p →0, Z→ 1, and the gas behaves as an ideal gas. • It is more accurate to express the compression factor as a power series of the pressure:

28. Virial Equation • Since The virial equation may be expressed as: • B'is known as the secondvirial coefficient. It is the largest correction term, making it the most important measure of nonideality. • C'is known as the thirdvirial coefficient. • D'is known as the fourthvirial coefficient. • The firstvirial coefficient is 1.

29. Virial Equation • According to Boyle’s Law, the pressure and volume of a gas are inversely proportional: • The compression factor, which is directly proportional to the pressure, is inversely proportional to the molar volume, Vm • As Vm→, Z→1, and the gas behaves as an ideal gas. • B is a temperature-dependent constant for a specific gas. • Expressing the compression factor as a power series of the molar volume:

30. Virial Equation • Since The virial equation may be expressed as: • The virial coefficientsB, C, and Dare not the same as B', C', andD' . • The virial coefficients are related however:

31. Boyle Temperature • For an ideal gas, since the compression factor Z = 1, the slope • For real gases: • and: • Asp →0,

32. Boyle Temperature • The second virial coefficient B' is temperature-dependent • B' < 0 at low temperatures • B' > 0 at high temperatures • The temperature at which B'(or B) is 0 is referred to as the Boyle temperature • Real gas behaves as an ideal gas

33. Gas Condensation • At constant temperature, a gas confined in a piston may be compressed so much that it begins to liquefy, or condense. • Consider the compression of one mole of CO2 at 20C, as represented by the p-V isotherm: • Initially (at A) the pressure increases as the volume decreases. At B, the pressure is large and the gas is no longer ideal. • At C, there is no further increase in pressure (represented by the horizontal line CDE). The gas is liquefying. The piston can move without additional resistance. • The pressure at which both liquid and gas are present at equilibrium is known as the vapor pressure.

34. Critical Constants • Consider the compression of one mole of CO2 at 31.04C, as represented by the p-V isotherm: • As the gas is compressed, no condensation occurs, no matter how much the gas is compressed. • The horizontal parts of the isotherms, which represent liquid-vapor equilibria have merged into a single point, known as the critical point. • The temperature at the critical point is known as the critical temperature, TC. • Liquid phase does not form above the critical temperature. • Critical constants – the temperature, pressure, and molar volume at the critical point.

35. Supercritical Fluids • Supercritical fluid – A single phase that fills the entire volume (like a gas) when the temperature and pressure are above the critical values, which may be more dense than the gas. • Consider the compression of one mole of CO2 at 40C, as represented by the p-V isotherm: • There is a point in which decreasing the volume has a small effect on the pressure. • The gas is behaving more like a liquid.

36. van der Waals Equation • The ideal gas equation is based upon the model that: • Gases are composed of particles so small compared to the volume of the gas that they can be considered to be zero-volume points in space. • There are no interactions, attractive or repulsive, between the individual gas particles. • In 1873, Johannes H. van der Waals (1837-1923), a Dutch physicist, developed an approximateequation of state for real gases that takes these factors into account: • A semiempirical equation, based upon experimental evidence, as well as thermodynamic arguments. • Awarded 1910 Nobel Prize in Physics for his work.

37. van der Waals Equation • Repulsive interactions between particles are taken into account by assuming that the particles behave as small, impenetrable hard spheres. • Assume that there are N molecules of gas in a volume V, each having a volume β. The actual volume for the gas molecules to occupy is: • If b is the volume per mole of gas molecules: • The actual volume for the gas molecules to occupy is:

38. van der Waals Equation • Correcting for the volume of the particles into the ideal gas equation: • Since • Taking into account the actual volume of the gas particle results in an increase in pressure relative to that predicted by the ideal gas law.

39. van der Waals Equation • The pressure of the gas also depends on: • The frequency of collisions of the gas particles with the walls of the container. • The force with which the particles strike the container. • Both are reduced by attractive forces between particles. • The attractive forces act with a strength proportional to the number density of the gas molecules in the container:

40. van der Waals Equation • The pressure is thus reduced by the attractive forces in proportion to the square of the number density. Taking this into account, the corrected pressure is: which is known as the van der Waals equation. • In terms of the molar volume, Vm, the van der Waals equation is:

41. van der Waals Equation • The van der Waals constants: • a • Pressure correction • Represents the magnitude of attractive forces between gas particles • Does not specify any physical origin to these forces • b • Volume correction • Related to the size of the particles • These constants: • Are unique to each type of gas. • Are not related to any specific molecular properties.

42. van der Waals Equation • Unlike the virial equation, which fits the behavior of real gases to a mathematical equation, the van der Waals equation is a mathematical model that attempts to predict the behavior of a gas in terms of real physical phenomena. • As a relatively simple model, the van der Waals equation is often not as accurate as the virial equation.

43. van der Waals Loops • van der Waals loops • Anomalous oscillations which occur for temperatures below the critical temperature. • Suggest that the pressure increases by increasing the volume. • Violate Boyle’s Law. • Replaced by horizontal lines in a procedure known as Maxwell constructions.

44. van der Waals Isotherms • The van der Waals equation Generates ideal gas isotherms at high temperatures and at large molar volumes. • At high temperature, the first term may be much greater than the second term. • At large molar volumes, and the ideal gas law is achieved:

45. Critical ConstantsFor the van der Waals Equation • The critical constants are related to the van der Waals coefficients. • The critical constants are found by solving for the critical point, which occurs with both the slope and curvature of the van der Waals isotherms are zero. • The slope is found from taking the first derivative of the pressure with respect to the molar volume: The curvature is found from taking the second derivative of the pressure with respect to the molar volume:

46. Critical ConstantsFor the van der Waals Equation • Solving these two equations in two unknowns (temperature and molar volume) gives the critical temperature and critical molar volume: • The critical pressure may be calculated by substituting the expressions for the critical temperature and critical molar volume into the van der Waals equation:

47. Critical Compression Factor, Zc • The critical compression factor, Zc, is defined as: • Substituting the expressions for the critical pressure, critical temperature, and critical molar volume for a van der Waals gas: • Since the critical compression factor, Zc, does not depend on the van der Waals parameters a and b, it should be the same for all gases. Experimentally, Zc, is found to be equal to 0.30.

48. Principle of Corresponding States • Because the van der Waals equation is only an approximation, it is useful to have a common scale on which properties of different gases can be compared. • Because the critical constants are characteristic properties of gases, they serve as a useful scale to compare different gases. • The reduced variables of a gas are determined by dividing the actual variable by the corresponding critical constant: • The reduced compression factor, Zr , may be defined as:

49. Principle of Corresponding States • van der Waals conducted an experiment in which two different real gases were confined to the same reduced volume and same reduced temperature. He noticed that the two different gases had approximately the same reduced pressure. • Principle of Corresponding States • “Different gases have the same reduced compression factors if they have the same reduced variables.” • “Real gases that have the same reduced volume and reduced temperature have the same reduced pressure.” • Allows for the determining the behavior of one gas by using the properties of another. • Only an approximation. It works best for gases composed of spherical molecules. Polar molecules show large deviations.

50. Principle of Corresponding States • Consider the dependence of the compression factor on the reduced pressure for four different gases at different reduced temperatures. • The use of reduced variables organizes the data onto single curves.