physics 2c lecture 5 chapter 21 - university of california

18
Chapter 21 Thermodynamic processes Specific Heat of an ideal gas Equipartition Theorem Physics 2c Lecture 5

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Page 1: Physics 2c Lecture 5 Chapter 21 - University of California

Chapter 21

Thermodynamic processes Specific Heat of an ideal gas

Equipartition Theorem

Physics 2c Lecture 5

Page 2: Physics 2c Lecture 5 Chapter 21 - University of California

PV diagram and Work

Page 3: Physics 2c Lecture 5 Chapter 21 - University of California

Work in cyclic processes

•  You have two identical cycles in the PV diagram.

•  One is traversed clockwise, the other counter-clockwise.

•  For which of the two does the gas do more work?

(a)  The clockwise one (b)  The counterclockwise one (c)  Neither, because in both cases the work done by

the gas is the same.

Page 4: Physics 2c Lecture 5 Chapter 21 - University of California

Favorite 4 types of processes

•  Isothermal: T=const dU=0 •  Isobaric: P=const •  Isochore: V=const W=0 •  Adiabat: dQ=0

This covers all the players in the game!

Page 5: Physics 2c Lecture 5 Chapter 21 - University of California

4 Processes

Page 6: Physics 2c Lecture 5 Chapter 21 - University of California

Isothermal (dU=0)

!

P =nRTV

Q =W = nRT dVVVi

V f

" = nRT lnVf

Vi

Page 7: Physics 2c Lecture 5 Chapter 21 - University of California

Isochoric (dV=0)

•  Constant volume => W=0 •  => Q=dU •  Define “Molar specific heat at

constant volume”:

!

Cv =Qn"T

=1n"U"T

Page 8: Physics 2c Lecture 5 Chapter 21 - University of California

Specific Heat for Ideal Monoatomic Gas

•  U = N <Ekin> •  From ideal Gas: <Ekin> = 3/2 kT ⇒ U = 3/2 N k T = 3/2 nRT

Use this to calculate specific heat:

Cv =1n!U!T

=32R

Page 9: Physics 2c Lecture 5 Chapter 21 - University of California

Isobaric (dP=0)

•  W,Q, and dU are all non-zero!

•  Define “molar specific heat at constant pressure”:

Q = !U + P!VQ = nCv!T + P!V = nCv!T + nR!T

!

Cp =Qn"T

= Cv + R

Page 10: Physics 2c Lecture 5 Chapter 21 - University of California

Molar Specific Heat of a mono-atomic Ideal Gas at

constant pressure

CP = Cv + R =52R

Page 11: Physics 2c Lecture 5 Chapter 21 - University of California

Adiabatic (Q=0)

•  No heat exchange with environment

•  Define “adiabatic exponent”:

!

"U +W = 0

!

" =Cp

CV

Page 12: Physics 2c Lecture 5 Chapter 21 - University of California

Adiabatic exponent of a mono-atomic Ideal Gas

! =Cp

Cv

=Cv + RCv

=53

Page 13: Physics 2c Lecture 5 Chapter 21 - University of California

Equipartition theorem

When a system is in thermodynamic equilibrium, the average energy per

molecule is 1/2 kT for each degree of freedom.

Page 14: Physics 2c Lecture 5 Chapter 21 - University of California

Quantum Effects

•  Not all degrees of freedom are accessible at low temperatures!

•  => Cv thus changes as temperature increases and degrees of freedom “unfreeze”.

•  At low T: only translational dof=3. •  At medium T: also rotational dof=5. •  At high T: also rotations & oscillations dof=7.

Page 15: Physics 2c Lecture 5 Chapter 21 - University of California

Note on di-atomic gases:

•  At high temperatures internal energy U is not just in the kinetic energy of the molecules!

•  E.g. H2 – Below 100K -> only 3 dof – 100K-3000K -> 5 dof , i.e. add 2 rotations – Above 3000K -> 7 dof, i.e. add Ekin & Epot for

vibration.

Page 16: Physics 2c Lecture 5 Chapter 21 - University of California

Adiabat in PV diagram

!

PV " = const

TV " #1 = constThis is derived in Chapter 21 of the book, the “Math Toolbox” prior to example 21-3. I expect you to be able to use this in problems !!!

Page 17: Physics 2c Lecture 5 Chapter 21 - University of California

The 4 Processes

Page 18: Physics 2c Lecture 5 Chapter 21 - University of California

You need to be able to analyze cyclic processes involving any

of these 4 processes!

E.g. Diesel & Otto engine Diesel: adiabat, isobar, adiabat, isochor Otto: adiabat, isochor, adiabat, isochor

E.g. I describe two paths in the PV plane, and you tell me which one requires more heat,

or does more work. E.g.: Problem 61,62 and cumulative problem 1 in Chapter 22.