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

(s1400 kPa – s1000 kPa) / (1400-1000) = -(v100 C – v60

C) / (100-60)

-1.005 X 10-4 = -1.0095 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 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

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)

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

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

Equations of State

Ideal gas law:

Pv = RT Van der Waals

(P + (a/v2))(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 cP

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

Mayer Relation

cP - cV = Tv2/ Known as the Mayer relation

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)

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