Transcript Short Version : 18. Heat, Work, & First Law of

Short Version : 18. Heat, Work, & First Law of Thermodynamics
18.1. The 1st Law of Thermodynamics
PE of falling weight
 KE of paddle
 Heat in water
Either heating or stirring can raise T of the water.
1st Law of Thermodynamics:
Increase in internal energy = Heat added  Work done
U  Q  W
dU dQ dW


dt
dt
dt
Thermodynamic state variable
= variable independent of history.
e.g., U, T, P, V, …
Not Q, W, …
Joule’s apparatus
18.2. Thermodynamic Processes
Quasi-static process:
Arbitrarily slow process such that system always
stays arbitrarily close to thermodynamic equilibrium.
Reversible process:
Any changes induced by the process in the universe (system
+ environment) can be removed by retracing its path.
Twater = Tgas & rises slowly
Reversible processes must be quasi-static.
Irreversible process:
Part or whole of process is not reversible.
e.g., any processes involving
friction, free expansion of gas ….
system always in
thermodynamic equilibrium
Work & Volume Changes
面積
W  Fext x
W  F x  p A x  p V
V2
W   d W   p dV
V1
Work done by gas on piston
Isothermal Processes
Isothermal process: T = constant.
V2
V2
V1
V1
W   p dV  
W  n R T ln
U
n RT
dV  n R T ln V
V
V2
V1
3
N kT
2
Q  W  n R T ln

V2
V1
U  0  Q  W
Isothermal processes
on ideal gas
V2
V1
Constant-Volume Processes & Specific Heat
Constant-volume process ( isometric, isochoric, isovolumic ) : V = constant
V  0
W  p V  0

U  Q
CV = molar specific heat at constant volume

U  Q  n CV T
Ideal gas: U = U(T)
Non-ideal gas:

CV 
1  dQ 
n  d T V
isometric processes
Uideal gas  n CV T
U  n CV T
for all processes
only for isometric processes
Isobaric Processes & Specific Heat
Isobaric Process : constant P
Isotherms
W  p  V2 V1

 p V
Q  U  W  U  p V
CP = molar specific heat at constant pressure
CP 
Q  n CP T
isobaric processes
n CP T  n CV T  p V
Ideal gas, isobaric :
 n CV T  n R T

CP  CV  R
1  dQ 
n  d T  P
Ideal gas
Adiabatic Processes
Adiabatic process: Q = constant
e.g., insulated system, quick changes like combustion, …
U  W
Tactics 18.1. 
W
p1 V1  p2 V2
 1
Prob. 62
cdf
p V   const
adiabat,
ideal gas
T V  1  const
Prob. 66

Adiabatic: larger p
CP
1
CV
TACTIC 18.1. Adiabatic Equation
dU  n CV dT
Ideal gas, any process:
Adiabatic process:
pV  n RT

dU   dW   p dV  n CV dT


p dV  V d p  n R dT
p
p dV  V dp
dV 
CV
R
 R  CV  p dV  CV V d p  0
Cp p dV  CV V d p  0
dV d p


0
V
p
 ln V  ln p  const  ln p V 


Cp
CV
p V   const
Example 18.3. Diesel Power
Fuel ignites in a diesel engine from the heat of compression (no spark plug needed).
Compression is fast enough to be adiabatic.
If the ignit temperature is 500C, what compression ratio Vmax / Vmin is needed?
Air’s specific heat ratio is  = 1.4, & before the compression the air is at 20 C.
T V  1  const
1 /   1
Vmax  Tmin 


Vmin  Tmax 
1 / 1.4 1
 273 K  500 K 


273
K

20
K


 11
Ideal Gas Processes
Cyclic Processes
Cyclic Process : system returns to same thermodynamic state periodically.
Example 18.4. Finding the Work
An ideal gas with  = 1.4 occupies 4.0 L at 300 K & 100 kPa pressure.
It’s compressed adiabatically to ¼ of original volume,
then cooled at constant V back to 300 K,
& finally allowed to expand isothermally to its original V.
How much work is done on the gas?
AB (adiabatic):
WAB 
WAB
BC (isometric):
CA (isothermal):
work done by gas:
p A VA  pB VB
 1
V 
pB  p A  A 
 VB 

 1
pA VA   VA  
100 kPa  4.0 L  1  41.41 

1     

 741 J
  1   VB  
1.4  1


WBC  0
WCA  n R T ln
VA
VC
 pA VA ln 4  555 J
WABCA  WAB  WBC  WCA  186 J
18.3. Specific Heats of an Ideal Gas
Ideal gas:
K 
1
m v2
2
3
 N kT
2

3
kT
2
3
U ideal gas
 n RT
2
5
CP  CV  R  R
2


1 U
3
 R
n T
2
5
C
 1.67
 P 
3
CV
CV 
Experimental values ( room T ):
For monatomic gases,   5/3, e.g., He, Ne, Ar, ….
For diatomic gases,   7/5 = 1.4, CV = 5R/2, e.g., H2 , O2 , N2 , ….
For tri-atomic gases,   1.3, CV = 3.4R, e.g., SO2 , NO2 , ….
Degrees of freedom (DoF) = number of independent
coordinates required to describe the system
Single atom: DoF = 3 (transl)
For low T ( vib modes not active ) :
Rigid diatomic molecule : DoF = 5 (3 transl + 2 rot)
Rigid triatomic molecule : DoF = 6 (3 transl + 3 rot)
The Equipartition Theorem
Equipartition theorem ( kinetic energy version):
For a system in thermodynamic equilibrium, each degree of freedom of a
rigid molecule contributes ½ kT to its average energy.
Equipartition theorem ( general version):
For a system in thermodynamic equilibrium, each degree of freedom described
by a quadratic term in the energy contributes ½ kT to its average energy.
f
  f n RT
U  n NA  k T 
2
 2
f

CP    1  R
2

f
CV  R
2
DoF ( f )
CV
CP

Monatomic
3
3/2
5/2
5/3
Diatomic
5
5/2
7/2
7/5
Triatomic
6
3
4
4/3

f 2
f
Example 18.5. Gas Mixture
A gas mixture consists of 2.0 mol of oxygen (O2) & 1.0 mol of Argon (Ar).
Find the volume specific heat of the mixture.
U O2 
5
nO R T
2 2
U Ar 
3
nAr R T
2
3
5

U mix   nO2  nAr  R T
2
2

Cmix
1 U mix

n T
 2.2 R
5
3
nO2  nAr
2
2
R
nO2  nAr
5
3
  2.0 mol    1.0 mol 
2
2
R
2.0 mol  1.0 mol
Quantum Effects
Quantum effect:
Each mechanism has a threshold energy.
Etransl < Erot < Evib
rotation+Translation+vibration
rotation+Translation
Translation
CV of H2 gas as function of T.
Below 20 K hydrogen is liquid,
above 3200 K it dissociates into individual atoms.
Reprise
Quasi-static process :
Arbitrarily slow process such that system always stays arbitrarily close to thermodynamic equilibrium.
Reversible process:
Any changes induced by the process in the universe (system + environment) can be removed
by retracing its path.
Dissipative work: Work done on system without changing its configuration, irreversible.
Insulated gas
a c : Free expansion with no dissipative work.
c b : Adiabatic.
a d : Adiabatic.
d b : Free expansion with no dissipative work.
a e : Adiabatic.
e b : Adiabatic dissipative work.
1st law:
The net adiabatic work done in all 3 processes are
equal (shaded areas are equal).