Real Gases

1. Real Gases A convenient measure of the deviation of a real gas from ideal behavior is given by the compressibilty factor, Z: Z  P V / n R T What is Z for an ideal gas and how does it vary with pressure? Note that at low pressures the compressibility factor depends linearly on the pressure: Z = 1 + (constant) P How does this result agree and/or disagree with the behavior observed for real gases? 3.1

2. The ideal gas equation of state can be improved by removing the restriction that the gas molecules or atoms have zero volume. Note that in the limit as temperature goes to zero on the Kelvn scale the molar volume of an ideal gas also goes to zero: lim V = R T / P ---> 0 T-->0 Why is this an absurd result for a real gas? We can improve this result by adding a constant term to the molar volume: lim V = R T / P + b ---> ? T-->0 The term b is roughly a measure of what molecular property? Note that for this “improved equation of state” Z depends linearly on P: Z = 1 + ( b / R T ) P 3.2

3. This “improved equation of state” can be further improved by removing the 2nd ideal gas restriction and accounting for the attractive forces that the molecules and atoms exert on each other. Since the attractive interactions between the molecules or atoms in the bulk fluid would decrease the momentum transferred in collisions with the container walls, these interactions decrease the pressure exerted by the gas. Since the molecules or atoms are mutually attracting each other, the reduction in pressure is proportional to the square of the concentration of the molecules or atoms: 3.3

4. Incorporating both of these improvements on the ideal gas law yields the van der Waals equation of state: P = n R T / (V- n b) - a C2 = n R T / (V - n b) - a n2 / V2 The proportionality constant a is a measure of the strength of the intermolecular or interatomic attractions. How should the proportionality contstant a vary with the number of electrons in the molecule or atom? The van der Waals equation of state is an example of a two parameter equation of state, where the parameters a and b are determined empirically by fitting experimental pressure, volume, temperature (PVT) data to the equation. What does the van der Waals equation reduce to in the limit of low pressure and high temperature? The equation, while modeling gas behavior more precisely than the ideal gas equation, is less general than the ideal gas equation in that a and b are different for different gases. Another drawback of the van der Waals equation, when compared to the ideal gas equation is that the equation is cubic in the volume, the number of moles, and their ratio, the molar volume. 3.4

5. Johannes Diderik van der Waals Johannes Diderik van der Waals was born November 23th, 1837 in Leiden as the eldest son in a family with eight children. Initially J.D. studied to be an elementary school teacher and taught school between 1856 and 1861. While studying to be a Head Master he attended lectures on Mathematics, Physics and Astronomy at Leiden University and starting in 1866 was engaged by a secondary school in The Hague as teacher in Physics and Mathematics. After seven years he became their subsitute director and in 1877 director. In 1873 he graduated from Leiden University. In his thesis Over de continuïteit van de gas- en vloeistoftoestand (On the continuity of the gaseous and liquid states) he published the well-known law, which is named after him. P = n R T / (V - n b) - a n2 / V2 This law is a correction on the law of ideal gases. It considers the own volume of the gas molecules and assumes a force between these molecules. Today these forces are known as "Van-der-Waals-forces". With this law, the existence of condensation and the critical temperature of gases could be predicted. 3.5

6. In 1877 J.D. becomes the first professor of Physics at the University "Illustre" in Amsterdam. Van der Waals was an excellent theoretical physicist and in Leiden he provided theoretical help to Kamerlingh Onnes in his successful attempt to liquify gasses. Van der Waals made a number of important contributions to the science of physics. In 1880 he formulated his "law of the corresponding states", in 1893 he developed a theory for capillary phenomena, and in 1891 a theory for the behaviour of mixtures of two materials. From 1875 to 1895 he was a member of the "Koninklijke Academie van Wetenschappen" ( Dutch Royal Academy of Science). In November of 1895 he and Kamerlingh Onnes became the first physicists to be honoured with the golden medal of the Genootschap ter bevordering van natuur-, genees- en heelkunde (Society for the benefit of physics, medical science and surgery) at Amsterdam. He is one of only twelve foreign members of the Academie des Sciences in Paris. In 1910 he received the Nobel prize in Physics joining Kamerlingh Onnes, Zeeman, Lorentz and Van 't Hoff as the fifth Dutch physisist to receive this honour. J.D. van der Waals died march 8th, 1923 at the age of 85. This material is taken from the WEB site http://www.phys.tue.nl/vdwaals/index-ns.htm (of course you have to read Dutch to understand it) maintained by a student society named after J. D. van der Waals at Technische Universisteit Eindhoven. 3.6

7. Summarizing the van der Waals equation of state is: P = n R T / (V - n b) - a n2 / V2 Some other well known two parameter equations of state are: the Peng-Robinson EQS: P = n R T / (V - n b) - a n2 / (V2 - n2 b2) Peng and Robinson are Chemical Engineers whose work on the phase behavior of fluids of interest to the Petroleum Industry has led to the formation of the DB Robinson Group of Companies that provides quality phase behavior and fluid property technology to the petroleum and petrochemical industries http://www.dbra.com/ the Redlich-Kwong EQS: P = n R T / (V - n b) - a n2 / (V(V + n b) T1/2) and the Berthelot EQS: P = n R T / (V - n b) - a n2 / (V2 T) What do these equations have in common? Are the constants a and b in the above equations all the same? Are all of the above EQSs also cubic equations in the molar volume? 3.7

8. While real gas equations of state can model liquids as well as gases, PVT data for condensed phases is often expressed in terms of isobaric coefficients of thermal expansion or in terms of isothermal compressibilities. Thecubical isobaric coefficient of thermal expansion, a,measures the fractional change in volume of some substance as the temperature on the substance is changed isobarically: a + (1/V) (V/T)p This equation can be viewed as an equation of state for a condensed phase at constant pressure. The cubical isobaric coefficient of thermal expansion for liquid water is: a = 2.5721x10-4 K-1 The partial differential equation defining a is easily solved by separating variables and integrating: VoV dV/V = ToT a dT The lower limits on these integrals represent a temperature To at which the volume Vo of the substance could be determined. Could you determine V o, if you knew the density, r o, at T o? The upper limits have been left unspecified to give after integration a function describing how the volume varies with temperature: ln (V/ Vo) = + a (T - To) Solving for V we get: V = Vo e + a (T -To)= Vo e - a Toe + a T 3.8

9. An approximate, but often encountered, form of this relation can be obtained by expanding the exponential as a power series and ignoring, since a is generally quite small, terms that are quadratic and higher in the temperature: V @ Vo [1 + a (T - To)] = Vo [1 + a DT] Can you derive an equation relating the density of a condensed phase to temperature? Using the following density versus temperature data for liquid Hg: - 10.0 oC 13.6202 g/mL + 15.0 oC 13.5585 g/mL - 5.0 oC 13.6078 g/mL + 20.0 oC 13.5462 g/mL - 0.0 oC 13.5955 g/mL + 25.0 oC 13.5340 g/mL + 5.0 oC 13.5832 g/mL + 30.0 oC 13.5217 g/mL + 10.0 oC 13.5708 g/mL + 35.0 oC 13.5095 g/mL determine a value for the cubical coefficient of thermal expansion of liquid Hg in units of K-1. The linear coefficient of thermal expansion of fused quartz glass is: a L = 5.5x10-7 K-1 = (1/L) (L/T)p What is the cubical coefficient of thermal expansion of this glass? 3.9

10. A barometer calibrated at 15.0 oC correctly reads the atmospheric pressure of 620.0 mm Hg when it is used at 15.0 oC. What will this barometer read when the atmospheric pressure is 620 mm Hg and the temperature is 30.0 oC? The linear isobaric coefficient of thermal expansion of Hg is 5.46x10-4 K-1. Ignore the thermal expansion of the glass. A thermocline marks a region of rapidly changing temperature. In the oceans the thermocline occurs at ~ 1000 meters below the ocean’s surface and denotes the depth where warm rapidly mixing surface waters transition to deeper denser more slowly mixing ocean waters at ~ 4 oC. By how many centimeters would the elevation of the ocean surface increase, if global warming resulted in an 1 oC increase in the average temperature of the ocean’s surface water. Take the average radius of the earth to be 6371.315 km. The cubical isobaric coefficient of thermal expansion for liquid water is a = 2.5721x10-4 K-1. 3.10

11. The isothermal compressibility, b,measures the fractional change in volume of some substance as the pressure on the substance is changed isothermally: b - (1/V) (V/P)T The isothermal compressibility for liquid water is: b = 4.46555x10-5 atm-1 Why does the equation defining isothermal compressibility have a minus sign? Can you separate varibles and integrate this equation to obtain a function describing how the volume of a substance will vary with pressure at constant temperature? A stainless steel “bomb” is filled with liquid water at 25.0 oC and 1.000 atm. This “bomb” is then heated to 50.0oC. The volume of the stainless steel “bomb” is assumed to remain constant during the heating. What will be the pressure inside the “bomb” at 50.0oC? Use the definitions of coefficient of themal expansion and isothermal compressibility and the “circle rule”: (V/P)T (P/T)V (T/V)P = - 1 (note how the variables are permuted among the derivatives in the “circle rule”) to derive a an equation relating pressure to temperature at constant volume. You will need the coefficient of thermal expansion and isothermal compressibility for liquid water found in the preceding notesto complete this problem. 3.11

12. Thermal Behavior of ZrW2O8 Bonus Problem A Bonus Problem dealing with the Thermal Behavior of ZrW2O8can be accessed by clicking the above link.