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Thermodynamics and Statistical Mechanics Equations of State

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Thermo & Stat Mech - Spring 2006 Class 22 Thermodynamic quantities Internal energy (U): the energy of atoms or molecules that does not give macroscopic motion. Temperature (T): a measure of the internal energy of a system. Heat (Q): a way to change internal energy, besides work. (Energy in transit.)

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Thermo & Stat Mech - Spring 2006 Class 23 Laws of Thermodynamics First law: đQ – đW = dU Q – W = U Energy is conserved

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Thermo & Stat Mech - Spring 2006 Class 24 Work done by a gas

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Thermo & Stat Mech - Spring 2006 Class 25 Work done by a gas

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Thermo & Stat Mech - Spring 2006 Class 26 Work done by a gas

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Thermo & Stat Mech - Spring 2006 Class 27 Configuration Work Product of intensive variable times corresponding extensive variable: đ W = xdY Gas, Liquid, Solid:PdV Magnetic Material: BdM Dielectric Material:EdP

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Thermo & Stat Mech - Spring 2006 Class 28 Equation of State For an ideal gas: PV = nRT P = pressure (N/m²)(or Pa) V = volume (m³) n = number of moles T = temperature (K) R = gas constant (8.31 J/(K·mole))

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Thermo & Stat Mech - Spring 2006 Class 29 Ideal gas law Ideal gas law:PV = nRT In terms of molar volume, v = V/n, this becomes: Pv = RT, or P = RT/v

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Thermo & Stat Mech - Spring 2006 Class 210 Real Substance

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Thermo & Stat Mech - Spring 2006 Class 211 Real Substance

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Thermo & Stat Mech - Spring 2006 Class 212 van der Waals equation of state v cannot be decreased indefinitely, so replace v by v – b. Then, Next account for intermolecular attraction which will reduce pressure as molecules are forced closer together. This term is proportional to v -2

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Thermo & Stat Mech - Spring 2006 Class 213 van der Waals equation of state This equation has a critical value of T which suggests a phase change. The next slide shows graphs for several values of T.

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Thermo & Stat Mech - Spring 2006 Class 214 van der Waals equation of state

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Thermo & Stat Mech - Spring 2006 Class 215 Critical Values

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Thermo & Stat Mech - Spring 2006 Class 216 van der Waals equation of state This can be expressed in term of dimensionless coordinates, P ', v ', and T ' with the following Substitutions:

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Thermo & Stat Mech - Spring 2006 Class 217 van der Waals equation of state This can also be written,

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Thermo & Stat Mech - Spring 2006 Class 218 Thermal Expansion Expansivity or Coefficient of Volume Expansion,.

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Thermo & Stat Mech - Spring 2006 Class 219 Thermal Expansion Usually, is positive. An exception is water in the temperature range between 0° C and 4° C. Range of is about: for gasses for solids.

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Thermo & Stat Mech - Spring 2006 Class 220 Linear Expansion Coefficient of Linear Expansion,.

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Thermo & Stat Mech - Spring 2006 Class 221 Relationship Between and

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Thermo & Stat Mech - Spring 2006 Class 222 Compressibility Volume also depends on pressure. Isothermal Compressibility:

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Thermo & Stat Mech - Spring 2006 Class 223 Bulk Modulus

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Thermo & Stat Mech - Spring 2006 Class 224 A Little Calculus

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Thermo & Stat Mech - Spring 2006 Class 225 Cyclical Relation

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Thermo & Stat Mech - Spring 2006 Class 226 Application Suppose you need:

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Thermo & Stat Mech - Spring 2006 Class 227 Application

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