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5.60 Thermodynamics & Kinetics Spring 2008

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5.60 Spring 2008

Lecture #4

H ≡ U + pV

H(T,p)

Enthalpy

page 1

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, T 1, V 1)

const . p

U1

gas (p , T2, V 2)

U2

∆U = q + w = q p − p ∆V ∆U + p ∆V = q p

define as H

∆U + ∆ ( pV ) = qp

H ≡ U + pV Choose

⇒ H (T , p )

∂H ⎞ What are ⎛⎜ ⎟

⎝ ∂T ⎠ p

•

⎛ ∂H ⎞ ⎜∂ ⎟ ⎝ T ⎠p

⇒

∆H = q p

∆ (U + pV ) = q p for a reversible constant p process

⎛ ∂H ⎞ ⎛ ∂H ⎞ dH = ⎜ dT + ⎜ ⎟ dp ⎟ ⎝ ∂T ⎠p ⎝ ∂p ⎠T ⎛ ∂H ⎞ and ⎜ ⎟ ? ⎝ ∂p ⎠T ⇒

⇒

for a reversible process at constant p (dp = 0)

dH = đq p ⇒

∂H ⎞ and dH = ⎛⎜ ⎟ dT ⎝ ∂T ⎠p

∂H ⎞ đq p = ⎜⎛ ⎟ dT ⎝ ∂T ⎠ p

∴

but

⎛ ∂H ⎞ ⎜ ⎟ = Cp ⎝ ∂T ⎠ p

đ q 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 , T 1)

− pV w = pV 1 1 2 2

⇒ ∴

=

gas (p , T 2)

∆ U = q + w = pV − p2V2 = −∆ ( pV ) 1 1 ∆ U + ∆ ( pV ) = 0 ⇒ ∆ (U + pV ) = 0 ∴

∆H = 0

Joule-Thomson is a constant Enthalpy process. ⎛ ∂H ⎞ ⎟ dp ⎝ ∂ p ⎠T

dH = C pdT + ⎜

⎛ ∂H ⎞ ⎜ ⎟ = −C p ⎝ ∂p ⎠T

⇒

Define

∴

⎛ ∂H ⎜⎜ ⎝ ∂p

⎛ ∂H ⎞ ⎟ dpH ⎝ ∂p ⎠ T

⇒

⎛ ∂T ⎞ ⎜ ⎟ ⎝ ∂p ⎠ H

Cp dT = − ⎜

← can measure this

⎛ ∆T ⎞ ⎜ ⎟ ⎝ ∆p ⎠ H

⎛ ∆T ⎞ ⎛ ∂T ⎞ lim ⎜ =⎜ ⎟ ⎟ ≡ µJT ← Joule-Thomson Coefficient ∆p →0 ⎝ ∆p ⎠ H ⎝ ∂ p ⎠ H ⎞ ⎟⎟ = −C p µJT ⎠T

and

dH = C pdT − 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 depends on T, no p dependence ⎛ ∂H ⎞ ⎟ = µJT = 0 for an ideal gas ⎜ ⎝ ∂p ⎠T

For a van der Waals gas:

⎛ ∂H ⎜⎜ ⎝ ∂p

⎞ a ⎟⎟ ≈ b − RT ⎠T 1.

If

⇒

a

so

then ⎜⎜

when T =T inv=

a Rb

a =T Rb inv

if

∆p < 0

(p2 < p1 )

then ∆T > 0

gas heats up upon expansion. 2.

If

a >b RT

⎛∆ ⎞ then ⎜⎜ T ⎟⎟ > 0 ⎝ ∆ p ⎠H

⇒

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

Proof that

Lecture #4

Cp = C V + R

page 4

for an ideal gas

⎛ ∂H ⎞ ⎟ , ∂ T ⎠p ⎝

⎛ ∂U ⎞ ⎟ ⎝ ∂T ⎠V

Cp = ⎜

CV = ⎜ H =U + pV ,  

pV = RT

↓ ↑     ⎛ ∂H ⎞ ⎛ ∂U ⎞ ⎛ ∂V ⎞ ⎜ ⎟ =⎜ ⎟ +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

⎞ ⎟ ⎟ ⎠...