Lecture 2 Copy

Dr. Ibrahim Al-Ansari 2/19/14 CHM 241 Spring 2014

Real Gases

•  have non-zero volume at low T and high P •  have repulsive and attractive forces between molecules

short range, important at high P

longer range, important at moderate P

At low pressure, molecular volume and intermolecular forces can often be neglected, i.e. properties à ideal.

21 B CPV RTV V

⎡ ⎤= + + +⎢ ⎥⎣ ⎦K

21PV RT B P C P′ ′⎡ ⎤= + + +⎣ ⎦K

mVV Vn

= =

virial Equations

B is the second second virial coefficient. C is the third virial coefficient. They are temperature dependent.

Van del Waals Equation

( )2aP V b RTV

⎛ ⎞+ − =⎜ ⎟⎝ ⎠


Compessibility Factor also known as compression factor

ideal

PV VZRT V

= =

Z

P0

1.0

2.0

NH3

H2

C2H4 CH4

Z

P0

1.0

2.0

0

1.0

2.0

NH3

H2

C2H4 CH4

Z

P

1.0

T1T2T3

Z

P

1.0

T1T2T3

The curve for each gas becomes more ideal as T à ∞


The van der Waals Equation 1

( )2aP V b RTV

⎛ ⎞+ − =⎜ ⎟⎝ ⎠

Intermolecular attraction = “internal pressure”

“molecular volume” ≈ excluded volume

( )3 34 23 32 / 2rπ = πσ

The initial slope depends on a, b and T:

2RT aPV b V

PV V aZRT V b RTV

= −−

= = −−

= 1+ 1

RTb− a

RT⎛⎝⎜

⎞⎠⎟

P + a

RT( )3 2b− aRT

⎛⎝⎜

⎞⎠⎟

P2 +…

1 ...T

Z abP RT RT∂⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠

⇒

(do the algebra)

/b a RT> molecular size dominant

forces dominant

Boyle Temperature

•  positive for

•  negative for

•  zero at

/b a RT</T a Rb=

~ ideal behaviour over wide range of P


condensation of Gases

Real gases condenses … don’t they?

supercritical fluid

gas liquid

P

P1

Pc

V

P2

Vc

T1

Tc T2

Tc, Pc and Vc are the critical constants of the gas.

Above the critical temperature the gas and liquid phases are continuous, i.e. there is no interface.


The van der Waals Equation 2

∂P∂V

⎛⎝⎜

⎞⎠⎟ T

= − RT

V − b( )2 +2aV 3 = 0

∂2 P∂V 2

⎛⎝⎜

⎞⎠⎟ T

= 2RT

V − b( )3 −6aV 4 = 0

The van der Waals Equation is not exact, only a model. a and b are empirical constant.

3 2 0RT a abV b V VP P P

⎛ ⎞− + + − =⎜ ⎟⎝ ⎠P

Vb 0

The cubic form of the equation predicts 3 solutions

Pc =a

27b2 Vc = 3b Tc =8a

27Rb

Zc =PcVc

RTc

= 38

TB = aRb

= 278

Tc

There is a point of inflection at the critical point, so…

slope:

curvature:

⇒

2RT aPV b V

= −−


The Principle of Corresponding States

Reduced variables are dimensionless variables expressed as fractions of the critical constants:

r r rc c c

P V TP V TP V T

= = =

Real gases in the same state of reduced volume and reduced temperature exert approximately the same reduced pressure.

They are in corresponding states.

If the van der Waals Equation is written in reduced variables,

( )r r r2r

3 3 1 8P V TV

⎛ ⎞+ − =⎜ ⎟

⎝ ⎠

Since this is independent of a and b, all gases follow the same curve (approximately).

Z

Pr

1.0 Tr = 1.5

Tr = 1.2

Tr = 1.0


Partial Differentiation for functions of more than one variable: f=f(x, y, …)

For a simultaneous increase

Take area as an example A xy=

For an increase

For an increase

y constant

x constant 1

2

in , in , x x A y xy y A x y

Δ Δ = ΔΔ Δ = Δ

ΔA = x + Δx( ) y + Δy( )− xy

= yΔx + xΔy + ΔxΔy

=ΔA1

ΔxΔx +

ΔA2

ΔyΔy + ΔxΔy

In the limits 0, 0x yΔ → Δ →

ΔA→ dA = ∂A∂x

⎛⎝⎜

⎞⎠⎟ y

dx + ∂A∂y

⎛⎝⎜

⎞⎠⎟ x

dy

∂f∂x

⎛⎝⎜

⎞⎠⎟ y

= limδx→0

f x + δx, y( )− f (x, y)δx

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

for a real single-value function f of two independent variables,

partial differential total differential

A xy=

2AΔ

1AΔ

Δx____

x

y


Partial Derivative Relations

y x

z zdz dx dyx y∂ ∂⎛ ⎞⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

( , , ) 0 ( ,, )f x y z z z x y= =Consider so

•  Partial derivatives can be taken in any order . 2 2z zx y y x∂ ∂=∂ ∂ ∂ ∂ yx y x

z zx y y x

⎡ ⎤⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎣ ⎦ ⎣ ⎦

•  Taking the inverse: 1

y y

z xx z

−⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

•  To find the patial partial derivative:

dz = 0 ⇒ ∂z∂y

⎛⎝⎜

⎞⎠⎟ x

dy = − ∂z∂x

⎛⎝⎜

⎞⎠⎟ y

dx

∂x∂y

⎛⎝⎜

⎞⎠⎟ z

= −∂z / ∂y( )x

∂z / ∂x( )y

= − ∂z∂y

⎛⎝⎜

⎞⎠⎟ x

∂x∂z

⎛⎝⎜

⎞⎠⎟ y

•  Chain Rule:

and

∂x∂y

⎛⎝⎜

⎞⎠⎟ z

∂y∂z

⎛⎝⎜

⎞⎠⎟ x

∂z∂x

⎛⎝⎜

⎞⎠⎟ y

= −1

∂y∂x

⎛⎝⎜

⎞⎠⎟ z

∂x∂z

⎛⎝⎜

⎞⎠⎟ y

∂z∂y

⎛⎝⎜

⎞⎠⎟ x

= −1


Partial Derivatives in Thermodynamics From the generalized equation of state for a closed system,

1P V T

V T PT P V

∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( ), , 0f P V T =

Six partial derivatives can be written:

but given the three inverses, e.g

and the chain rule

1

P P

V TT V

−∂ ∂⎡ ⎤⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

there are only two independent “basic properties of matter”. By convention these are chosen to be:

the coefficient of expansion expansion (isobaric), and

The third derivative is simply

1P

VV T

∂⎛ ⎞α = ⎜ ⎟∂⎝ ⎠

1T

VV P

∂⎛ ⎞κ = − ⎜ ⎟∂⎝ ⎠

( )( )

//

P

V T

V TPT V P

∂ ∂∂ α⎛ ⎞ = − =⎜ ⎟∂ ∂ ∂ κ⎝ ⎠

the coefficient of isothermal cmpressibility.

∂V∂T

⎛⎝⎜

⎞⎠⎟ p

∂T∂P

⎛⎝⎜

⎞⎠⎟ v

∂P∂V

⎛⎝⎜

⎞⎠⎟ T

∂T∂V

⎛⎝⎜

⎞⎠⎟ P

∂P∂T

⎛⎝⎜

⎞⎠⎟ v

∂V∂P

⎛⎝⎜

⎞⎠⎟ T


The Euler Relation

Suppose δz = A x, y( )dx + B x, y( )dy

Is z an exact differential, i.e. dz?

dz is exact provided yx

A By x

∂ ∂⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠cross-differentiation

because then

A = ∂z∂x

⎛⎝⎜

⎞⎠⎟ y

∂A∂y

⎛⎝⎜

⎞⎠⎟ x

= ∂2z∂y∂x

B = ∂z∂y

⎛⎝⎜

⎞⎠⎟ x

∂B∂x

⎛⎝⎜

⎞⎠⎟ y

= ∂2z∂x∂y

The corollary also holds (if exact, the above relations hold).

State functions have exact differentials.

Path functions do not.

New thermodynamic relations may be derived from the Euler relation.

dU = TdS − PdV∂T∂V

⎛⎝⎜

⎞⎠⎟ S

= − ∂P∂S

⎛⎝⎜

⎞⎠⎟V

e.g. given that

it follows that

∂

δ

Lecture 2 Copy

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