# handout04 - 2.16 The Virial Equation of State AIM Introduce...

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2.16 The Virial Equation of State AIM: Introduce an additional Equation of State for gasses. NOTE: The term virial, due to Clausius, comes from the Latin word, vis , for ’force’. The Virial EoS is also called the Virial Expansion. REMINDER: The central idea behind the Van der Waals EoS is that both δ p and the interparticle attractive force are proportional to the density ( n V , see page 2–31). IDEA: We can get an even better EoS by introducing a power series in n V : pV nRT = Z (2.84) = 1 + B 2 ( T ) n V + B 3 ( T ) n V 2 + . . . , (2.85) where Z is called the compression factor , B 2 ( T ) and B 3 ( T ) are functions of temperature T and are called the second and third virial coefficients . NOTE: In principle, the Virial EoS can be made as accurate as needed by adding terms. In practice, terms larger than 3 rarely used. QUESTION: How can we determine virial coefficients? ANSWER: experiment; Measure many p - V - T data points of a gas. 2–43

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Fit the data points to obtain the virial coefficients. theory (Not in this class); can be calculated from Statistical Mechanics under the assumption of a particular interparticle interaction function such as the Lennard-Jones interaction potential. NOTE: The virial EoS can be very accurate; but it is not based on a microscopic model like the VdW EoS. 2.17 Coexistence of Phases and Critical Points REMINDER: We have seen that below the critical temperature T c , the isotherm is not monotone, indicating that a phase change can occur. Near phase change boundaries, different phases can coexist. If we plot the points of a substance that exist dependent on pressure, volume and temperature, we obtain the PVT-surface. For example, for a substance that contracts on freezing: 2–44
NOTE: The triple line is a line of constant T and P in which solid, gas and fluid phases coexist. Viewed in the P-T plane, this line becomes the triple point . NOTE: It is possible to freeze a fluid below its normal freezing point without it turning into ice: super cooling. This can happen when there are no seeds around which crystals can form. The fluid then can suddenly freeze when it is disturbed. 2–45

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EXAMPLE: Freezing rain: super-cooled droplets form in the clouds; freeze upon impact with objects on the ground. EXAMPLE: Slushies. Freeze carbonated drink. Upon opening of the bottle, ice crystals form. See [ ] See [ ] 2–46
2.18 The Boltzmann Distribution AIM: Introduce the all important Boltzmann distribution and show where it comes from. QUESTION: Why is the Boltzmann distribution so important? ANSWER: It is the central ’mathematical bridge’ between the microscopic and macroscopic world of Thermodynamics.

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