Zeroth Law of Thermodynamics and Equilibrium of State

Zeroth Law of Thermodynamics

and Equilibrium of State

Matter

Universe

Energy

Physical Chemistry

Physical chemistry – understanding the quantitative aspects of chemical phenomena.

Thermodynamics deals with the properties of macroscopic systems at equilibrium and measures of changes.

Observations of macroscopic behavior of matter have established the laws of thermodynamics.

The energy of the universe is constant.

The entropy of the universe increases.

1.1 State of a System

SYSTEM + SURROUNDINGS = UNIVERSE

BOUNDARY

System part of the physical universe under consideration The state of a system is denoted with state

functions (or state variables).

Systems

open closed isolated

Systems in Equilibrium

Homogeneoussystem – a single phase with properties uniform throughout

Heterogeneous system – more than one phase

Equilibrium a state with properties independent of time and space (no fluxes)

State at equilibrium is defined entirely by the state variables not by the history of the system

PATH Independent

State Function

= f ( )

STATE STATE Functions

T, V, P, S, E, H, A, G, n, etc.

Intensive variables – independent of the size of the system pressure, temperature, density, …

Extensive variables – depend on the size of the system

volume, mass, internal energy, entropy, …

Degrees of Freedom

F

D

N

N

s p

2 F = N − N +

p s

2

D = + F N = N +

The number of variables required to describe the state of a system is a generalization of experimental observations, i.e., empirical.

1.2 The Zeroth Law of Thermodynamcis

A

B C

A

B C

A

B C

Upon thermal contact between two objects at different temperature, thermal energy transfers from the object at higher temperature to the other object at lower temperature until the same thermal state (

thermal equilibrium

If system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other.

Temperature is the state function of system denoting its thermal state.

Temperature Scale

A relation between the thermodynamic properties of a substance at equilibrium

Charles and Gay-Lussac found that the volume of a gas varies linearly with temperature at specified pressure.

Temperature scale

P

V

Θ

Θ

Θ

Isotherms for a fluid

2 2

1 2

1 1

( , ) P V

PV = Θ Θ φ

2 2 2

1 1 1

P V T

PV = T

2 1

P V PV T = T

PV nR

T =

The equation of state for an ideal gas

1.3 The Ideal Gas Temperature Scale

0

lim

P

T PV

→

R

=

The unit of thermodynamic temperature,

1 kelvin or 1 K

At 1 atm (101,325 Pa), the freezing point of water is 273.15K and the boiling point is 373.12K. The Celsius scale is defined by

t/℃ = T/K – 273.15

Plot of versus temperature for a given amount of real gas at two low pressure P₁and P₂

V

Pressure

Force on unit area

Under the standard condition 0°C, 1atm, the height of Hg column is 76.00cm

/

f mg m

P gh gh

A V h V ρ

= = = =

g = 9.80665 m/s² and ρ_Hg= 13.5951 X10³ kg/m³

1 atm = (13.5951 X10³kg/m³) (9.80665 m/s² ) (0.76m )

= 101,325 N/m² = 101,325 Pa = 1.01325 bar

1 Pa = 1 kg / ms² 1 bar = 10⁵Pa

1 atm = 760 torr = 760 mmHg (at 0°C) = 1.01325 X10⁵ Pa 1 atm = 14.6960 psi (lb / in²)

Pressure Units

The Gas Constant

At STP (0 ^oC, 273.15 K, 1 atm), the molar volume of an ideal gas is 22.414 L.

(1 atm)(22.414 L)

0.082058 L atm / mol K (1 mol)(273.15K)

R PV

= nT = =

The recommended value based on measurements of the speed of sound in argon (1986)

R = 8.31451 J / mol K

The universal gas constant R

= 1.98722 cal / mol •K

= 0.082057 L•atm / mol •K

1.5 Real Gases

Real gases behave like ideal gases only in the limit of low pressures and high temperatures.

The compressibility factor is a convenient measure of the deviation from ideal gas.

Z PV

= RT

Influence of high pressure on the compressibility factor at 298K

Influence of pressure on the

compressibility factor for nitrogen at different temperatures.

The Virial Equation of State

In 1901, H. Kamerlingh-Onnes proposed

The coefficient B and C are the second and third virial coefficients, respectively. Virial means “force”

in Latin.

1

PV B C

Z RT V V

= = + + +

Gas B

10^-6m³/mol

C 10^-12m⁶/mol²

H₂ 14.1 350

He 11.8 121

N₂ –4.5 1100

O₂ –16.1 1200

Ar –15.8 1160

CO –8.6 1550

Second and third virial coefficient at 298K

Second virial coefficient B

The Boyle Temperature

The temperature at which the secont virial coefficient B is zero.

Gas T_B/ K

He-4 22.64

H₂ 110.04

N₂ 327.22

O₂ 405.88

CO₂ 714.81

NH₃ 995

CH₄ 509.66

C₂H₄ 624 At the Boyle temperature ( B = 0 ), a gas

behaves nearly ideally over a range of pressures.

The curvature at high pressures depends on the sign of the third virial coefficient.

1.6 P-V-T Surface for a One-Component System

P V − − T

tthe triple point

1.7 Critical Phenomena

 A

tie line

 ① - ② - ③ - ④ No meniscus conversion

 Supercritical fluid

 At the critical point,

0

T Tc

P V

 ∂  =

 ∂ 

 

2

2

0

T Tc

P V

 ∂  =

 

 ∂ 

 The

isothermal compressibility

1

T

V V P κ _{= − } ^{ ∂ } _

 ∂ 

 The large differences,

spontaneous

1.8 The van der Waals Equation

The ideal gas equation is based on point particles that do not interact except in elastic collisions.

Real gas molecules are not point particles (less volume) and interact each other (more compressible).

The van der Waals equation of states (1877)

2

( )

P a V b RT V

 

+ − =

 

 

Gas a

L² bar / mol²

b

L / mol Gas a

L² bar / mol²

b L / mol

H₂ 0.2476 0.02661 CH₄ 2.283 0.04278

He 0.03457 0.02370 C₂H₆ 5.562 0.06380

N₂ 1.408 0.03913 C₃H₈ 8.779 0.08445

O₂ 1.378 0.03183 n-C₄H₁₀ 14.66 0.1226

Cl₂ 6.579 0.05622 iso-C₄H₁₀ 13.04 0.1142

NO 1.358 0.02789 n-C₅H₁₂ 19.26 0.1460

NO 5.354 0.04424 CO 1.505 0.03985

The van der Waals Equation

van der Waals isotherms

The Maxwell construction

Area(ABC) = Area(CDE) B, D

0

T

P V

 ∂  =

 ∂ 

 

B – D : unstable

0

T

P V

 ∂  〉

 ∂ 

 

A – B, D – E : metastable

The van der Waals Equation

lim

( )

V

P a V b PV RT

→∞

V

 

+ − = =

 

 

2

( )

P a V b RT V

 

+ − =

 

 

The compressibility factor for a van der Waals gas

1 1 /

PV V a a

Z = RT = V b − RTV = b V − RTV

− −

and using

/ 1

b V  1

1 1 x x

x = + + +

− 

1

1 a b

Z b

RT V V

   

= +   −   +     + 

Apply the ideal gas relation for only the second term to obtain,

1 1 a

Z b P

RT RT

 

= +   −   + 

The Boyle temperature B

T a

= bR

The van der Waals Equation

Example 1.9

(0.083145 L bar / mol K)(350 K)

0.416 L/mol 70 bar

V RT

= P = =

(a)

(b)

RT a

P V b V

= −

−

(0.08315)(350) 5.562 70 V 0.06380 V

= −

−

Using Mathematica,

Solve[70 == .08315*350/(v-.0638)-5.562/(v^2),v]

{{v -> 0.124906 -0.0804012 i},{v -> 0.124906 +0.0804012 i},{v -> 0.229738}}

0.2297 L/mol

V =

Euler’s Theorem

A homogeneous function of degree k

f ( λ λ x

, x

,  , λ x

) = λ

f x x ( ,

,  , x

)

Euler’s theorem ^{1 2}

1

( , , , )

j i

N

N i

i i x x

k f x x x x f

=

x

 ∂ 

= ∑   ∂  



1

1 2 1 2 1 2

( , , ,

) ( , , ,

) ( , , ,

) V λ λ n n  λ n = λ V n n  n = λ V n n  n

 All intensive properties are homogeneous of degree 0.

 All extensive properties are homogeneous of degree 1.

0

1 2 1 2 1 2

( , , ,

) ( , , ,

) ( , , ,

)

T λ λ n n  λ n = λ T n n  n = T n n  n

Partial Molar Volume

Euler’s theorem ^{1 2}

1

( , , , )

j i

N

N i

i i x x

k f x x x x f

=

x

 ∂ 

= ∑   ∂  



1 2 1 1 2 2

1 , , _j 2 , , _j , , _j

N N N

T P n T P n N T P n

V V V

V n n n V n V n V n

n n n

 

 ∂   ∂  ∂

=   ∂   +   ∂   + +    ∂   = + + + 

Partial molar volume

, ,{ _{j i}} i

i T P n

V V

n

 ∂ 

=   ∂  

 is the change in V when an infinitesimal amount (dn_i) is added to the solution at constant T, P, and all other n_j.

i

d

V n

1 1 2 2

d V = V n d + V n d + +  V

d n

V

 is the change in V when 1 mol of i is added to an infinitely large amount of the solution at constant T, P.

= + + + 