Most likely macrostate the system will find itself in is the one with the maximum number of microstates.
E1
1(E1)
E2
2(E2)
Total microstates =
Ξ© (πΈ1,πΈ2 )=Ξ©1(πΈ1)Ξ©2(πΈ2)
To maximize :
E1
1(E1)
E2
2(E2)
Most likely macrostate the system will find itself in is the one with the maximum number of microstates.
E1
1(E1)
E2
2(E2)TkdE
d
dE
d
B
1lnln
2
2
1
1
Using this definition of temperature we need to describe real systems
Boltzmann Factor (canonical ensemble)
TkBeP
)(
π β² (οΏ½βοΏ½ )ππ£π₯ππ£ π¦ππ£ π§βπβ ππ£2
2ππ΅π ππ£ π₯ππ£ π¦ππ£π§
π β² (οΏ½βοΏ½ )ππ£π₯ππ£ π¦ππ£ π§βπβπ(π£π₯
2+π£π¦2+π£ π§
2 )2ππ΅π ππ£ π₯ππ£ π¦ππ£π§
π β² (οΏ½βοΏ½ )ππ£π₯ππ£ π¦ππ£ π§βπβππ£ π₯
2
2ππ΅π ππ£ π₯πβππ£π¦
2
2ππ΅π ππ£ π¦πβππ£π§
2
2ππ΅π ππ£ π§
βπ (π£ π₯ )ππ£π₯
Integrating over the two angular variables we can get the probability that the speed of a particle is between and :
π β² (οΏ½βοΏ½ )π£2 sinπ ππ£ππ ππβπβ ππ£2
2ππ΅π π£2sin π ππ£ππππ
β π (π£ )ππ£βπβ ππ£2
2ππ΅π π£2 ππ£
For to be a proper probability distribution/density function:
β«0
β
π (π£ )ππ£=1
β π (π£ )ππ£= 4βπ ( π
2ππ΅π )3 /2
π£2πβ ππ£2
2ππ΅π ππ£
Maxwell-Boltzmann speed distribution
0 10 20 30 40 50 60 70 80 90 1000
0.02
0.04
0.06
0.08
0.1
0.12
T = 10
0 10 20 30 40 50 60 70 80 90 1000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
T = 100
0 10 20 30 40 50 60 70 80 90 1000
0.002
0.004
0.006
0.008
0.01
0.012
T = 1000
β π (π£ )ππ£= 4βπ ( π
2ππ΅π )3 /2
π£2πβ ππ£2
2ππ΅π ππ£
β¨π£ β©=β«0
β
π£π (π£ )ππ£=β 8ππ΅πππ
β¨π£2 β©=β«0
β
π£2 π (π£ )ππ£=3ππ΅ππ
=π£πππ 2
βππ=πππ£ 2 π (π£ )ππ£ sinπ cos2π ππ
The pressure on the wall due to all the particles in the gas is:
π=ππβ«0
β
π£2 π (π£)ππ£ β«0
π/2
sin π cos2πππ
ΒΏππ β¨π£2 β© 13
ΒΏππ3ππ΅ππ
13
ΒΏπππ΅π=ππππ΅π
βππ=πππ΅π
Only till to include only those particles hitting the wall from the left
Efficiency of a Carnot engine
ππ΄ ,π π΄ ,π h
ππ΅ ,π π΅ ,π h
ππΆ ,π πΆ ,π π
ππ· ,π π· ,π π
β Ξπh=π π h lnπ π΅
π π΄p
V
β Ξππ=π π π lnπ π·
π πΆ
β Ξπ=0βπhπ π΅
πΎβ1=π ππ πΆπΎβ1
β Ξπ=0βπ ππ π·
πΎβ1=πhπ π΄πΎβ1
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
Pr_b
v(1) v(2) v(3)
Calculates the relevant area for the Maxwell constructions(v(2)-v(1))*Pr_b - integral(Prfunc,v(1),v(2))
integral(Prfunc,v(2),v(3)) - (v(3)-v(2))*Pr_b
P (
bar)
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