General Relations
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Transcript of General Relations
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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
(s1400 kPa – s1000 kPa) / (1400-1000) = -(v100 C – v60
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 dv We can also write the entropy as a function of T and v
ds = (s/T)v dT + (s/v)T dv
We can end up with
du = cvdT + [T(P/ T)v – P]dv 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 = cpdT + [v - T(v/T)P]dP If we know u or h we can find the other from
the definition of hh = u + (Pv)
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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 dvds = (s/T)P dT + (s/P)T dP
ds = (cV/T) dT + (P/T)V dvds = (cP/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
(cv/v)T = T(P/T2)v
(cp/P)T = -T(v/T2)P
cP - cV = -T(v/T)P2 (P/v)T
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Equations of State
Ideal gas law:
Pv = RT Van der Waals
(P + (a/v2))(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 cP
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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: = 0eaT
The more you compress a liquid, the harder the
compression becomes
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Mayer Relation
cP - cV = Tv2/ Known as the Mayer relation
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Using Heat Capacity Equations
cP - cV = -T(v/T)P2 (P/v)T
cP - cV = Tv2/ 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