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5.60 Thermodynamics & Kinetics
Spring 2008
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5.60 Spring 2008
Lecture #3
Isothermal Gas Expansion
page 1
(∆T = 0)
gas (p1, V1, T) = gas (p2, V2, T)
Irreversibly (many ways possible)
(1)
Set pext = 0
p= 0
T
T
p= 0
p 1,V1
p 2,V2
v2
w (1) = − ∫ pext dV = 0
V1
(2)
Set pext = p2
p2
T
p2
T
p 2,V2
p 1,V1
v2
w (2) = − ∫ p2dV = −p2 (V2 −V1 )
V1
p
p1
p2
V1
-w(2)
V2
Note, work is negative: system expands against surroundings
5.60 Spring 2008
(3)
Lecture #3
page 2
Carry out change in two steps
gas (p1, V1, T) = gas (p3, V3, T) = gas (p2, V2, T)
p1 > p3 > p2
p2
p3
p3
T
T
T
p 2,V2
p 3,V3
p 1,V1
v3
v2
V1
V3
w (3) = − ∫ p3dV − ∫ p2dV = − p3 (V3 −V1 ) −p2 (V2 −V3 )
p
p1
More work delivered to
surroundings in this case.
p3
p2
V1 V3
V2
-w(3)
(4)
Reversible change
p = pext throughout
V
p
wrev = − ∫ 2 pdV
V1
p1
p2
V1
-
V2
rev
For ideal gas:
V
wrev = − ∫ 2
V1
Maximum work delivered to
surroundings for isothermal gas
expansion is obtained using a
reversible path
V
p
nRT
dV = −nRT ln 2 = nRT ln 2
V
V1
p1
5.60 Spring 2008
Lecture #3
page 3
The Internal Energy U
dU = d-q + d-w
(First Law)
dU = C pathdT − pext dV
And U (T ,V
)
⇒
⎛ ∂U ⎞
⎛ ∂U ⎞
dU = ⎜
⎟ dT + ⎜
⎟ dV
∂
T
∂
V
⎝
⎠V
⎝
⎠T
Some frequent constraints:
dU = d-qrev + d-wrev = d-qrev – pdV
Reversible
⇒
•
Isolated
⇒
d-q = d-w = 0
•
Adiabatic
⇒
d-q = 0
•
Constant V
⇒
•
(p = pext )
reversible
⇒ dU = d-w =
-pdV
w = 0 ⇒ dU = d-qV
Constant V
∂U ⎞
⎛ ∂U ⎞
dU = ⎛⎜
⎟ dT + ⎜
⎟ dV
⎝ ∂T ⎠V
⎝ ∂V ⎠T
but also
⎛ ∂U ⎞
⇒
d-qV = ⎜
⎟ dT
⎝ ∂T ⎠V
d-qV = CV dT
So
⇒
⎛ ∂U ⎞
⎜
⎟ = CV
⎝ ∂T ⎠V
very important result!!
⎛ ∂U ⎞
⎟
⎝ ∂V ⎠T
dU = CV dT + ⎜
dV
what is this?
5.60 Spring 2008
Lecture #3
⎛ ∂U ⎞
⎟ )
⎝ ∂V ⎠T
Joule Free Expansion of a Gas
gas
(to get ⎜
vac
gas (p1, T1, V1) = gas (p2, T2, V2)
Since q = w = 0
Recall
⇒
Adiabatic
q=0
Expansion into Vac.
(pext=0)
w=0
dU or ∆U = 0
Constant U
⎛ ∂U ⎞
⎟ dV = 0
⎝ ∂V ⎠T
dU = CV dT + ⎜
⎛ ∂U ⎞
⎜
⎟
⎝ ∂V ⎠T
dVU = −CV dTU
⎛ ∂U ⎞
⎜
⎟ = −CV
⎝ ∂V ⎠T
Joule did this.
•
page 4
⎛ ∂T ⎞
⎜
⎟
⎝ ∂V ⎠U
measure in Joule exp't!
⎛ ∆T ⎞
⎜
⎟
⎝ ∆V ⎠U
⎛ ∆T ⎞
⎛ ∂T ⎞
∴ dU = CV dT − CV ηJ dV
⎟ =⎜
⎟ ≡ ηJ
⎝ ∆V ⎠U ⎝ ∂V ⎠U
Joule coefficient
lim ⎜
∆V →0
For Ideal gas
⇒
ηJ = 0
dU = CV dT
U(T)
exactly
Always for ideal gas
only depends on T
The internal energy of an ideal gas depends only on temperature Consequences
⇒
⇒
∆U = 0
∆U = ∫ CVdT
For all isothermal expansions or
compressions of ideal gases
For any ideal gas change in state