MIT OpenCourseWare
http://ocw.mit.edu
5.60 Thermodynamics & Kinetics
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
5.60 Spring 2008
Lecture #4
Enthalpy
H(T,p)
page 1
H ≡ U + pV
Chemical reactions and biological processes usually take place under
constant pressure and with reversible pV work. Enthalpy turns out
to be an especially useful function of state under those conditions.
reversible
=
gas (p, T1, V1)
gas (p, T2, V2)
const . p
U1
U2
∆U = q + w = qp − p ∆V
∆U + p ∆V = qp
∆U + ∆ ( pV ) = qp
H ≡ U + pV
Choose
⇒
H (T ,p )
define as H
∆ (U + pV ) = qp
⇒
∆H = q p
for a reversible constant p process
⎛ ∂H ⎞
⎛ ∂H ⎞
⎟ dp
⎟ dT + ⎜
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
dH = ⎜
⇒
⎛ ∂H ⎞
∂H ⎞
What are ⎛⎜
⎟ ?
⎟ and ⎜
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
•
⎛ ∂H ⎞
⎜
⎟
⎝ ∂T ⎠ p
for a reversible process at constant p (dp = 0)
⇒
dH = đq p
⇒
⎛ ∂H ⎞
and dH = ⎜
⎟ dT
⎝ ∂T ⎠ p
∂H ⎞
⎟ dT
⎝ ∂T ⎠ p
đq p = ⎛⎜
∴
but
⎛ ∂H ⎞
⎜
⎟ = Cp
⎝ ∂T ⎠ p
đq p = C p dT
also
5.60 Spring 2008
•
Lecture #4
⎛ ∂H ⎞
⎜
⎟
⎝ ∂p ⎠T
⇒
page 2
Joule-Thomson expansion
adiabatic, q = 0
porous partition (throttle)
gas (p, T1)
w = pV
1 1 − p2V2
⇒
∴
=
gas (p, T2)
∆U = q + w = pV
1 1 − p2V2 = −∆ ( pV
∆U + ∆ ( pV ) = 0 ⇒
∴
)
∆ (U + pV ) = 0
∆H = 0
Joule-Thomson is a constant Enthalpy process.
⎛ ∂H ⎞
⎟ dp
⎝ ∂p ⎠T
dH = C pdT + ⎜
⇒
⎛ ∂H ⎞
⎛ ∂T ⎞
⎜
⎟ = −C p ⎜
⎟
⎝ ∂p ⎠T
⎝ ∂p ⎠H
Define
∴
⎛ ∂H
⎜⎜
⎝ ∂p
⎛ ∂H ⎞
⎟ dpH
⎝ ∂p ⎠T
C p dT = − ⎜
⇒
← can measure this
⎛ ∆T ⎞
⎜
⎟
⎝ ∆p ⎠ H
⎛ ∆T ⎞
⎛ ∂T ⎞
=⎜
lim ⎜
⎟
⎟ ≡ µJT ← Joule-Thomson Coefficient
∆p
→0
⎝ ∆p ⎠H ⎝ ∂p ⎠H
⎞
⎟⎟ = −C pµJT
⎠T
and
dH = C p dT − C p µJT dp
5.60 Spring 2008
Lecture #4
U(T),
For an ideal gas:
page 3
pV=nRT
H ≡ U (T ) + pV = U (T ) + nRT
H (T )
⇒
only
⎛ ∂H
⎜
⎝ ∂p
depends on T, no p dependence
⎞
⎟ = µJT = 0 for an ideal gas
⎠T
For a van der Waals gas:
⎛ ∂H
⎜⎜
⎝ ∂p
⎞
a
⎟⎟ ≈ b −
RT
⎠T
1.
If
⇒
a
<b
RT
µJT ≈
⇒
⎛ ∆T ⎞
⎟⎟ < 0
∆
p
⎝
⎠H
a
−b = 0
RT
T >
so
then ⎜⎜
when T =T inv=
a
Rb
a
=T
Rb inv
if
∆p < 0
then ∆T > 0
( p2 < p1 )
gas heats up upon expansion.
2.
If
a
>b
RT
⎛ ∆T ⎞
⎟⎟ > 0
⎝ ∆p ⎠H
then ⎜⎜
⇒
T <
so
a
=T
Rb inv
if
∆p < 0
then ∆T < 0
gas cools upon expansion.
Tinv >> 300K
⇒
for most real gases.
Use J-T expansion to liquefy gases
5.60 Spring 2008
Lecture #4
Proof that C p = CV + R
page 4
for an ideal gas
⎛ ∂H ⎞
⎟ ,
⎝ ∂T ⎠ p
⎛ ∂U ⎞
⎟
⎝ ∂T ⎠V
Cp = ⎜
CV = ⎜
H = U + pV ,
pV = RT
↓
↑
⎛ ∂H ⎞
⎛ ∂V ⎞
⎛ ∂U ⎞
⎜
⎟ =⎜
⎟
⎟ +p⎜
⎝ ∂T ⎠ p ⎝ ∂T ⎠ p
⎝ ∂T ⎠ p
↑
⎛R ⎞
⎛ ∂U ⎞ ⎛ ∂V ⎞
C p = CV + ⎜
⎟ + p ⎜⎜ ⎟⎟
⎟ ⎜
⎝ ∂V ⎠T ⎝ ∂T ⎠ p
⎝ p ⎠
=0 for ideal gas
∴
C p = CV + R