Chapter 2 The First LawChapter 2 The First Law

2.1 The basic concepts

Thermodynamics field of physics that describes and correlates the physical properties of macroscopic systems of matter and energy.

1.1.1 System and surrounding

System the parts of the world in which we have a special interest.

Surroundingswhere we make our measurement.

system water Open system

system water+ vapor

Closed system

system water+gas

Isolated system

adiabatic

2.1.2 Thermodynamic properties

Pressure p , volume temperature V , temperature T ,

internal energy U, enthalpy H , entropy S ……

extensive property extensive property = intensive property

m

VV

n

m

V e.g.

e.g mass, volume

Extensive property a property that depends on the amount of substance in the sample.

e.g. Temperature ， pressure ， density

Intensive property a property that is independent of the amount of substance in the sample.

2.1.3 Definition of phase

Phase a homogeneous part of a system.

Homogeneous system one phaseHeterogeneous system two or more phase, interface

2.1.4 The equilibrium state

Thermal equilibrium T1= T2 = T3=… = Tex

Mechanical equilibrium p1= p2 = p3=… = pex

Phase equilibrium = = =…

Chemical equilibrium A= B= Y = Z …

Thermodynamic state—— equilibrium state：

2.1.5 State and state function

The value of a state function depends only on the present state of the system and not on its past history.

The state of a macroscopic system in equilibrium can be described in terms of such measurable properties as T, p, and V, which are known as thermodynamic variables( or state functions).

State description Classical mechanics:Thermodynamics: p,V,T…

,r p

2.1.6 Process and path

Process When a macroscopic system moves from one state of equilibrium to another, a thermodynamic process is said to take place.

Path

(b)Isobaric process p1 = p2 ＝ pex

(a)Isothermal process T1 = T2 ＝ Tex

(c)Isochoric process V1 = V2

(d)Adiabatic process Q=0

(e)Cyclic process

1. p, V, T process

(f)expansion against constant pressure pex=constant

(g) free expansion pressure pex=0

2 . Phase transition process

aA + bB = yY + zZ

0 = ΣBB

nB,0 is the number of mole of substance B present at the start of the reaction .

3 . Chemical reaction process

B —stoichiometric number of B

nB = nB,0 nB =B ， d nB = B d

A= － a ， B= － b ， Y= y, Z= z

extent of reaction — , units is mol 。1 1

B B B Bd d Δ Δn n or

2.1.7 Work and heat

W>0 W is done on the system by the surroundings

W <0 system does work on its surrounding

Work The energy transfer between system and surroundings due

to a macroscopic force acting through a distance

Heat The energy transfer between system and surroundings due to a temperature difference

Q>0 when heat flows into the system from the surroundings

Q<0 when heat flows into the surroundings from system

2.1.8 Internal energy U

Type of workType of work ddWW

ExpansionExpansion

Surface expansionSurface expansion

ElectricalElectrical

--ppexexddVV

ddAA

ddqq

(2) U is an extensive property;

(1) U is state function ;

(3) the absolute value of U is unknown.

The total energy of system

2.2 The first law

U =U2 － U1 = Q + W closed

system dU ＝ δQ ＋ δW U＝ Q＋W

Isolated system

Cyclic process

Adiabatic process

Q=0, W=0 , U=0

U=0, Q= W

Q =0, U = W

Conservation of energy

It is impossible to built a first kind of perpetual motion machine.Work and heat dU = Q + W

2.2.1 Expansion work

ex ex exδ dz d dW F p A z p V

2

1exd

V

VW p V

1. Free expansion

pex=0, W=0

2

1su su 2 1

ex

d ( )V

VW p V p V V

p V

-

2.Expansion against constant pressure

3. Reversible expansion

Different expansion

Reversible expansion

W= - pexdV = - pdV

2

1r d

V

VW p V

Quasi-static process

Isothermal reversible expansion

Consider the isothermal, reversible expansion of a perfect gas:

2

1

2

1

2

1

2

1

d

d

d

ln

V

V

V

V

V

V

W p V

nRTV

VT

nR VVV

nRTV

Reversible process

One where the system is always infinitesimally close to equilibrium and an infinitesimal change in conditions can reverse the process to restore both system and surroundings to their initial state.

Characteristic

(a) Infinitesimally close to equilibrium.

(b) Tex ＝ T ； pex ＝ p.

(c) Both system and surroundings can be restore to their

initial state through reverse process.

2.2.2 Heat transaction

Consider closed system

dU = Q + W = Q + Wexp + W’

dV=0, W’=0 , dU = QV , U = QV

(a) Calorimetry

A constant-volume bomb calorimeter. The `bomb' is the central vessel, which is massive enough to withstand high pressures. The calorimeter is the entire assembly shown here. To ensure adiabaticity, the calorimeter is immersed in a water bath with a temperature continuously readjusted to that of the calorimeter at each stage of the combustion.

Definition of molar heat capacity

m

def 1( )

d

QC T

n T

(b) Heat capacity

Molar heat capacity at constant volume

,m

def ( ) δ1 1( )

dV V

VV

C T Q UC T

n n T n T

2

1

,m ( )d

T

V VTQ U nC T T

(c) Enthalpy

def H U pV

W ＝－ pexV

W ＝－ pexV, W′ ＝ 0 ， U = Qp － pexV

U2 － U1 = Qp － pex(V2 － V1)

p1 = p2 = pex

U2 － U1 = Qp － (p2 V2 － p1V1)

Qp = (U2 ＋ p2 V2) － (U1 ＋ p1V1) = (U ＋ pV)

Isobaric process

(1) H is state function ;

(2) H is an extensive property;

(3) the absolute value of U is unknown.

Enthalpy

For a closed system,

p=const . W’=0

Qp = H

δQp = dH

Cp ， m ＝ a ＋ bT ＋ cT 2 ＋ dT 3

or Cp ， m ＝ a ＋ bT ＋ c′T － 2

approximate empirical expression

2

1

,m ( )d

T

pTH nC T T

,m

( ) δ def 1 1( )

dp p

pp

C T Q HC T

n n T n T

Molar heat capacity at constant pressure

Perfect gas Cp,m-CV,m=R

2424

2.2.3 Adiabatic changes

dU ＝ CVdT, dU ＝ W

2 2

1 1

,m d d

T T

V VT TW U C T nC T if CV,m=const.

W ＝ U = n CV,m(T2 － T1)

(a) The work of adiabatic change

dU ＝ δW,

if δW′ ＝ 0 　 then CVdT ＝－ pexdV

Reversible process, pex ＝ p,

perfect gas ddV

VC T nRT

V

d d0

V

T nR V

T C V

perfect gas Cp － CV ＝ nRd d

0p V

V

C CT V

T C V

perfect gas γ=constant

(b) heat capacity ratioγ and adiabatsd d

( 1) 0T V

T V

def /p VC C

ln{ T } ＋ (γ1) ln{ V } = constant TVγ-1 = constant

pVT

nR constantγpV

nRTV

p (1 ) / constantTp

( perfect gas, reversible process, closed system, W′ ＝ 0 .)

Equations of adiabatic reversible process

2

1

d

V

VW p V

pV =constant

1

1 1 1

2

11

p V VW

V

1

1 1 2

1

11

p V pW

p

(c) Work of adiabatic reversible process of perfect gas

2.2.4 Phase transition

1. Enthalpies of phase transition

T=const. , p =const. ,W'=0 pQ H

vaporization :vapH m,

fusion : fusH m,

sublimation : subH m,

transition: trsH m

Enthalpy of

2. Expansion work of phase transition

at constant T and p

W ＝－ p(V － V)

β-gas phase,α-liquid phase (or solid phase)

V >>V, W － pVβ

β perfect gas W ＝－ pVβ ＝－ nRT

3. U of phase transition

U ＝ H － p(Vβ － Vα)

Vβ>>Vα, U ＝ H － pVβ

perfect gas U ＝ H － nRT

U ＝ Qp ＋ W

2.3.1 Molar enthalpies and internal energies of chemical change

BB

0 BaA + bB = yY + zZ

Br m

B

U UU

n

B

r mB

H HH

n

2.3 Thermochemistry

2.3.2 Standard enthalpy changes

mr m B

def ( ) (B, )H T H T ，yy

Standard molar enthalpies of the substance B at temperature T

and pressure p 。

(B=A, B, Y, Z; =phase state)

2.3.3 Standard states of substances

Standard pressure p＝ 100kPa

gas: p = p , T, perfect gas

solid or liquid : pex = p , T

rHm (T) = yHm

(Y, , T ) ＋ z Hm (Z, , T )

－ a Hm (A, , T ) － b Hm

(B, , T )

For reaction aA ＋ b B →yY ＋ zZ

Pure, unmixed reactant in their standard states Pure, separated products in their standard states

2.3.4 Relation between rHm and rUm

For a reaction B

BB0

rH m (T, liquid or solid) ≈rU m

(T, l or s)

rHm ( T) ＝ rUm (T) ＋ RT B (g)

rH m (T) ＝ BH m

(B, , T )

＝ BU m (B, ,T ) ＋ B[p V m (B,T)]

T= Constant V=constant W′ ＝ 0 QV ＝ rU

T= Constant p=constant W′ ＝ 0 Qp ＝ rH

2.3.5 Hess’s law

A Cr mΔ ( )H Ty

Br m,1Δ ( )H Ty

r m,2Δ ( )H Ty

rHm (T ) = rHm,1 (T ) ＋ rHm,2 (T )

The standard enthalpy of overall reaction is the sum of the standard enthalpies of the individual reactions into which a reaction may be divided.

rHm (T) in term of fHm

(B, ,T )

r Hm (T) = Bf Hm

(B, ,T )

reactants

elements

products

r Hm

Ent

halp

y,

H

2.3.6 Standard enthalpies of formation fHm (B, ,T )

r m B m

def ( ) (B,phase state ) H T H T ，yy

rHm (T)in term of cHm

(B, ,T )

r Hm (T) = – BcHm

(B, ,T ) *r m B m

def ( ) (B,phase state ) H T H T ，yy

reactants

CO2(g), H2O(l)

products

r Hm

Ent

halp

y, H

2.3.7 Standard molar enthalpies of combustion cHm

(B,phase state ,T )

2.3.8 The temperature dependence of reaction enthalpies

1H y2ΔH y

4ΔH y3ΔH y

r m 1Δ ( )H Ty

aA + bB yY + zZ

r m 2Δ ( )H Ty

aA + bB yY + zZ

r H m (T1) = H 1 ＋ H 2 ＋ r H m (T2) ＋ H 3 ＋ H 4

2

1

Br m 2 r m 1 ,m Δ ( ) Δ ( ) (B)d

T

pTH T H T C T y y

2

1

1 ,m Δ (A)d ,

T

pTH a C T y 2

1

2 ,m Δ (B)d ,

T

pTH b C T y

2 2

1 1

3 ,m 4 ,m Δ (Y)d , Δ (Z)d

T T

p pT TH y C T H z C T y y

B Cp,m(B)=yCp,m(Y) ＋ zCp,m(Z) - aCp,m(A)- bCp,m(B)

If T2 ＝ T,T1 ＝ 298.15K,

Kirchhoff’s law

2

Br m r m ,m 298.15KΔ ( ) Δ (298.15K) (B, )d

T

pH T H C T y y

2

1

Br m 2 r m 1 ,m Δ ( ) Δ ( ) (B)d

T

pTH T H T C T y y

2.4 State function and exact differentials

2.4.1. Exact differentials

d d dy x

Z Zz x y

x y

Z = f (x, y ),

2.4.2 Internal energy

d d d

d d

T V

VT

U UU V T

V T

UV C T

V

U = f (T, V ),

The Joule experimentT

U

V

＝０

Perfect gas dU=CVdT, or U=f(T)

2

1

Δ dT

VTU C T

2.4.3 Enthalpy

d d d d dppT T

H H HH p T p C T

p T p

H = f (T, p ),

dH=CpdT, or H=f(T) 2

1

Δ dT

VTU C T

Perfect gasT

H

p

＝０

2.4.4 The Joule-Thomson effect

Q ＝ 0

U ＝ W

or U2 － U1 ＝ p1V1 －

p2V2 U2 ＋ p2 V2 ＝ U1 ＋ p1

V1 H2 ＝ H1

W ＝ p1V1 － p2V2

Isenthalpic process U＝ f(T)

Joule-Thomson coefficient

J-T

def

H

T

p

p ＜ 0,

J-T ＜ 0, heating ；

J-T ＞ 0, cooling ；

J-T ＝ 0, T unchanged