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General Relations Thermodynamics Professor Lee Carkner Lecture 24
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PAL #23 Maxwell Determine a relation for ( s/ P) T for a gas whose equation of state is P( v -b) = RT ( s/ P) T = -( v / T) P P( v -b) = RT can be written, v = (RT/P) + b ( s/ P) T =-( v / T) P = -R/P
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PAL #23 Maxwell Verify the validity of ( s/ P) T = - ( v / T) P for refrigerant 134a at 80 C and 1.2 MPa Can write as ( s/ P) 80 C = -( v / T) 1.2 MPa Use values of T and P above and below 80 C and 1.2 MPa (s 1400 kPa – s 1000 kPa ) / (1400-1000) = -( v 100 C – v 60 C ) / (100-60) -1.005 X 10 -4 = -1.0095 X 10 -4
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Key Equations We can use the characteristic equations and Maxwell’s relations to find key relations involving: enthalpy specific heats so we can use an equation of state
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Internal Energy Equations du = ( u/ T) v dT + ( u/ v ) T d v We can also write the entropy as a function of T and v ds = ( s/ T) v dT + ( s/ v ) T d v We can end up with du = c v dT + [T( P/ T) v – P]d v This can be solved by using an equation of state to relate P, T and v and integrating
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Enthalpy dh = ( h/ T) v dT + ( h/ v ) T dv We can derive: dh = c p dT + [ v - T( v / T) P ]dP If we know u or h we can find the other from the definition of h h = u + (P v )
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Entropy Equations We can use the entropy equation to get equations that can be integrated with a equation of state: ds = ( s/ T) v dT + ( s/ v ) T dv ds = ( s/ T) P dT + ( s/ P) T dP ds = (c V /T) dT + ( P/ T) V d v ds = (c P /T) dT - ( v/ T) P dP
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Heat Capacity Equations We can use the entropy equations to find relations for the specific heats ( c v / v ) T = T( P/ T 2 ) v ( c p / P) T = -T( v / T 2 ) P c P - c V = -T( v / T) P 2 ( P/ v ) T
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Equations of State Ideal gas law: P v = RT Van der Waals (P + (a/ v 2 ))( v - b) = RT
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Volume Expansivity Need to find volume expansivity = For isotropic materials: = where L.E. is the linear expansivity: L.E. = Note that some materials are non-isotropic e.g.
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Volume Expansivity
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Variation of with T Rises sharply with T and then flattens out Similar to variations in c P
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Compressibility Need to find the isothermal compressibility == Unlike approaches a constant at 0 K Liquids generally have an exponential rise of with T: = 0 e aT The more you compress a liquid, the harder the compression becomes
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Mayer Relation c P - c V = T v 2 / Known as the Mayer relation
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Using Heat Capacity Equations c P - c V = -T( v / T) P 2 ( P/ v ) T c P - c V = T v 2 / Examples: Squares are always positive and pressure always decreases with v T = 0 (absolute zero)
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Next Time Final Exam, Thursday May 18, 9am Covers entire course Including Chapter 12 2 hours long Can use all three equation sheets plus tables
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