Series of transitions · This principle permits to determine which processes are spontaneous. Spontaneous change is a change that shows a natural tendency to occur without any external

Physical Chemistry EPM/03 1

A quote of the week

(or camel of the week):

Unprovided with original learning, unformed in the habits of thinking, unskilled in the arts of composition, I resolved to write a book.

Edward Gibbon


Series of transitions

Series of processes (transitions) in a given system is a number of tran-

sitions such as the final state of transition 1 (FS1) is also the initial

state of transition 2 (IS2), FS2 is IS 3, and so on.

Change in any state function X for a series of N transitions may be cal-

culated as:

but also as:1N ISFS

XXX −=∆ ∑∆=∆N

1iXX

We frequently use this fact to calculate ∆X for two states, replacing the

unknown overall transition (path) by a series of transitions for each leg

of which we know how to calculate its respective ∆Xi.

For quantities not being state fuctions, only

the second approach is applicable, e.g.: ∑=N

1iqq


Thermodynamic cycle

Thermodynamic cycle is a special series of processes, in which the final

state of last transition (FSN) is also the initial state of transition 1 (IS1).

Change in any state function X for a cycle is equal to 0. This is a simple

conclusion from the definition of the state function.

Quantities not being state fuctions must be calculated for the

cycle in the same manner as for any series of transitions.

0N

1

=∆=∆ ∑ iXX or 0=∫dX

One can even say that if a circular integral of certain function is

zero, then the function integrated is a state function.


Thermodynamic cycle (2)


Thermodynamic cycle (3)

This graph does not represent a cycle.

Shown are two ways of moving the

system from state 1 to state 3.

One leads directly from state 1 to state 3

in a single transition (probably an

isotherm), while the other achieves the

same in two legs: 1-2 and 2-3. According

to what was said before, changes in ∆U, ∆H (and all other state

functions) must be the same for both paths (1-3 and 1-2-3). It is

also clearly visible that w1-3 > w1-2-3, while q1-3<q1-2-3.


II law of thermodynamics

This principle permits to determine which processes are

spontaneous. Spontaneous change is a change that shows a

natural tendency to occur without any external influence.

First principle, for example, would permit suchphenomena like flow of heat from a cooler to a warmer

body or flow of gas from a container where lowerpressure exists to the one of higher pressure without anychanges in the system or its surrounding as long as the

rule of preservation of energy is obeyed.


II law of thermodynamics (2)

There are several statements expressing this principle.

Construction of a periodic (cyclic) engine

transforming heat into equivalent amount of work

without causing changes in the system or its

surroundings is impossible.

Engine contradicting the above statement is known

as a perpetuum mobile (second kind).



According to lord Kelvin (William Thomson) :

It is impossible to convert

heat completely into work

in a cyclic process

HEAT

WORK


According to Rudolf Clausius:

Heat generally cannot

spontaneously flow

from a material at

lower temperature to a

material at higher

temperature (this flow

being the exclusive

effect of a process).

high temperature

HEAT

low temperature



Entropy

2

1

2

1

T

T

q

q−= 0

2

2

1

1 =+T

q

T

q

We will not discuss the properties of Carnot’s cycle

here, but it can be proven that:

2

12

2

21

T

TT

q

qq −=

+

Every reversible cycle on the P – V plane may

be represented as a sum of certain number N of

Carnot’s cycles, hence:

0N

1

=∑i

i

T

q

This state function was later named ENTROPY.

∫ = 0T

dqand for N→∞


Entropy (2)

∫∩

−==

AB

AB ssT

dq

T

dqds and

ENTROPY is mathematically defined as:

Entropy of an isolated system in reversible processes:

T

dqds

syst

syst = T

dqds sur

sur = sursyst dqdq −=

0=+= sursyst dsdsds

Let’s consider an isolated system consisting of two closed systems

arranged in such a way that the second system constitutes

surroundings of the first system. Therefore:


Entropy (3)

0>+= sursyst dsdsds

It may be proven (from Carnot’s theorem) that for irreversible

processes:

If an irreversible process occurs in an isolated system, entropy of

the system increases.

If a reversible process occurs in an isolated system, entropy of

the system does not change.

0≥ds

Spontaneous processes are always irreversible!!!

Spontaneous processes are only these, which

increase the total entropy of the universe.

Generally:


Entropy (4)

Calculating changes in entropy for systems where no chemical

reactions occur:

T

dqdS =

In isobaric processes (quite analogous a way):

dUdq =T

dTCdS V=

∫=−=∆f

i

T

T

Vif dT

T

TCSSS

)(

T

dTCdS P=

In isochoric processes (assuming system contains 1 mole):

i

f

V

T

T

V

T

TCdT

T

CS

f

i

ln==∆ ∫

dHdq = ∫=∆f

i

T

T

P dTT

TCS

)(

i

f

PT

TCS ln=∆

if CV=const


Entropy (5)

Calculating changes in entropy for systems, where no chemical

reactions occur:

T

dqdS =

In reversible isothermal processes (assuming system contains 1 mole):

i

f

V

VR

T

w

T

qS ln=−==∆

Entropy is an extensive quantity, hence:

(this may be applied to all the formulae) Sns ∆⋅=∆

UNITS: entropy ∆s: molar entropy ∆S:

K

J

⋅molK

J


Entropy (6)

Calculating changes in entropy for systems where no chemical

reactions occur:

For any process leading from state characterized by Ti, Pi, Vi to

state Tf, Pf, Vf. (still assuming system contains 1 mole)

Under assumption CV=const

T

dqdS = PdVdUdq +=

V

dVR

T

dTC

TT

dUdS V +=+=

PdV

i

f

i

f

V

f

i

f

iV

V

VR

T

TC

V

dVR

T

dTCS lnln +=+=∆ ∫∫


Entropy as a measure of

disorder (1)

GAS



disorder (2)

LIQUID



disorder (3)

SOLID STATE



disorder (4)

SOLID STATE

GAS

LIQUID

order entropy

disorder

Ludwig Eduard Boltzmann

WkS ln⋅=



disorder (5)

WkS ln⋅=

where:

S - entropy of a system containg given amount of substance,

k - Boltzmann constant equal to 1.38×10-23 J/K (k=R/N

A, N

Ais Avogadro's number)

W - disorder of the substance (system), expressed as the num-

ber of ways the chemical species (atoms, molecules) may

be arranged while maintaining the same total energy.


Heating curve (1)

T=298K; H0

PT1 (melting)

PT2 (boiling)

P=const=1,013·105 Pa

∆H2

∆H1

TPT2=Tboil

TPT1=Tmel

PP CH

Ttg

1=

∂

∂=αGAS

LIQUID

SOLID STATE


Heating curve (2)

Absolute entropy

Tmelt Tboil

∆Sboil

∆Smelt

S(0)

∫

∫

∫

+

∆++

∆+

+=

T

T

gP

boil

boil

T

T

cP

melt

melt

TsP

boil

boil

melt

melt

T

dTTC

T

H

T

dTTC

T

H

T

dTTCSTS

)(

)(

)()0()(

,

,

0

,


Trouton’s rule

Standard molar entropy of vaporization of almost all liquids

is approximately the same and equal to

85 J·K-1·mol-1

When one mole of any liquid turns into its vapors (gas), it leads

to similar increase in disorder („amount of disorder”).

The rule is not obeyed when there are strong specific interactions

in liquids, e.g. in water (109,1 J·K-1·mol-1), where hydrogen

bonds exist. Also light gases do not fullfil the rule


III law of thermodynamics

Nernst theorem:

When temperature approaches absolute

zero, changes in entropy accompanying all

processes, both physical and chemical,

approach zero, too.

∆S →0, when T→0

Debye’s extrapolation:3

aTCP =

0lim0

=∆→TS Walther

Hermann Nernst


III law of thermodynamics

(2)

If entropy of all elements in their respective most stable forms is 0 at

T=0, then all substances must have positive entropy, which at T=0

may assume value S=0, and which actually assumes this value for all

perfectly crystalline substances (including chemical compounds).

For perfect substances

S(0)=0


Entropy of phase

transitions

Entropy of a phase transition at temperature

of this phase transition and at constant pressure.PT

PTPT

T

HS

∆=∆

Absolute entropy of elements and chemical compounds is

calculated according to the formula shown with heating curve (2)

including the relevant phase transitions up to 298 K and at

standard pressure.

Its values (per one mole of substance) may be found in tables

S0298


Entropy of reactions

Change in entropy accompanying any chemical rection (standard

entropy of reaction), may be calculated according to the formula:

One can notice, that this formula is analogous to the earlier ones

for ∆Ur ∆CP ∆Hr (always final minus initial state).

0

298,,1

0

298,,1

0

298, rei

n

iipri

n

iir SnSnS ∑∑

==

−=∆

Documents

Series of transitions · This principle permits to determine which processes are spontaneous. Spontaneous change is a change that shows a natural tendency to occur without any external