General Relations Thermodynamics Professor Lee Carkner Lecture 24

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

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 ) / ( ) = -( v 100 C – v 60 C ) / (100-60)  X = X 10 -4

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

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

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 )

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 

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

Equations of State   Ideal gas law: P v = RT  Van der Waals (P + (a/ v 2 ))( v - b) = RT

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.

Volume Expansivity

Variation of  with T   Rises sharply with T and then flattens out   Similar to variations in c P 

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

Mayer Relation  c P - c V = T v  2 /   Known as the Mayer relation

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) 

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