Thermodynamic variables

Path Function State Function

B

I

II

IIIIII WWW

A

III

(VB-VA)I = (VB-VA)II = (VB-VA)III

(state Function)

(Path Function)

Between two states the change in a state variable is always the same regardless of which path the system travels.

Differential of a state function is called

EXACT DIFFERENTIAL

a

b

a b VVdV

a

b

a b WWdW Differential of a path function is called

inexact differential.

I

II

x

a

b

xa xb

ya

yb

I area b

a

b

aydxdz

Path dependent hence inexact differential

II area I area

)(

aa

b

a

b

a bb

b

ayxyxxydxdyydxdz

Does not depend on path hence exact differential

Simple test to see whether a differential is exact

(Euler’s Criterion for exactness)

VTTV

TV

V

p

TT

p

dVV

pdT

T

pdp

VTfp

V

thatcondition With the

then

function state a is ),(

"Do not worry too much about your difficulties in   mathematics, I can assure you that mine are still greater.“

      -- Albert Einstein

Molar volume of an ideal gas

TP

m

p

m

T

mm

m

P

R

PP

R

P

RT

T

dTP

RdP

P

RTdV

dTT

VdP

P

VdV

P

RTV

22

2

When a system is at equilibrium, its state is defined entirely by the state variables, and not by the history of the system.

The properties of the system can be described by an equation of state which specifies the relationship between these variables.

The state of a system is defined as the complete set of all its properties which can change during various specified processes.

State variables are also called state properties.

State Variables

P,T,V

Cut into half

P,T,V/2 P,T,V/2

P and T are intensive variables, V is an extensive variable

The variables often form pairs such that their product has the dimensions of energy (e.g. pressure & volume in a gas). The intensive member plays the role of force and the extensive the role of displacement.

We can influence a system either by doing work (e.g. compression), or thermally (e.g. heat with a

flame).

Work is done when an object is moved against an opposing force.

The energy of the system is its capacity to do work.

Heat is energy in transit due to temperature difference.

When a system does work, it stimulates orderly motion in the surroundings.

Work is transfer of energy that make use of organized motion.

Heat is the transfer of energy that makes use of unorganized molecular motion.

When energy is transferred to the surroundings as heat, the transfer stimulates

disordered motion of the atoms in the surroundings.

Joule’s experiment

adiabatic

Joule heated water by performing work on it, in this case by rotating a paddle wheel.

He found that the temperature rise was dependent only on the amount of work but independent of how the work was performed (e.g. quickly or

slowly, large or small paddle wheel).

Temperature rise means there is change in state.

The property of the system whose change is calculated in this way is called internal energy.

adiabaticwU

Same change in state (temperature rise) can be achieved by allowing heat to flow in.

System in state 1

Heat reservoir

system Heat reservoir

Heat reservoir

System in state 2

done) work (no qU

Heat conducting Wall

Energy can be converted from one form to another, but it cannot be created or destroyed.

wqU

Change in internal energy of a closed system is equal to the energy that passes through its boundary as heat or work.

dwdqdU

state of changes malinfinitesi aFor

•The first law contains three essential features :

• It is a statement of the principal of conservation of energy.

• It requires the existence of the internal energy function.

• It leads to the definition of heat as energy in transit.

U is a function of the state variables and can be written as

),( TVfU

V

W

P

Q

or

But Q and W are not functions of the state so we cannot determine quantities like

For an adiabatic system,

dwdU

dq

0

This means that work needed to change an adiabatic system from one specified state to another specified state is a state function.

wad is a state function.ifad UUw

General expression for workdzd Fw

f

i

V

V ex

ex

ex

Vpw

Vpdw

zApw

d

d

dd

(a) Free expansion

pex=0, w=0(b) Expansion against constant pressure

(Expansion of gas formed in a chemical reaction)

h

P1,V1,TP2,V2,T

P1

P2

V1 V2

P1

P2

V1 V2

P1

P2

V1V2

P1

P2

V1V2

P1

P2

V1V2

For infinite number of step

Infinite state expansion is called reversible process.

These infinite-step processes at constant temperature are reversible because the energy accumulated in the surroundings in the expansion is exactly the amount required to compress the gas back to the initial state.

P1

P2

V1V2

h

P1,V1,TP2,V2,T

h

P1,V1,T

P1

P2

V1 V2

P2

P1

V2 V1

P2,V2,T

Important points about reversible processes

They can be reversed at any point in the process by making an infinitesimal change.

A reversible expansion or compression requires

•A balancing of internal and external pressure.

•Time to reestablish equilibrium after each infinitesimal step.

•Absence of friction.

For the processes in the chemical industry, the greater the irreversiblity, the greater is the loss in capacity to do work. Every irreversibility has its cost.

Ways to approach reversibility

Heat may be transferred nearly reversibly if the temperature gradient across which it is transferred is made very small.

Electrical charge may be transferred nearly reversibly from a battery if a potentiometer is used so that the difference in electric potential is very small.

A liquid may be vaporized nearly reversibly if the pressure of vapor is made only very slightly less than the equilibrium vapor pressure.

The variables often form pairs such that their product has the dimensions of energy). The intensive member plays the role of force and the extensive the role of displacement.

Type of Work Intensive variable

Extensive Variable

Differential work

Expansion P V -PdV

Surface (surface tension)

A dA

Elongation F L FdL

Electrical (potential diff.)

Q dQ

edwdwdqdU )kind PV(exp

Heat

Heat is energy in transit due to temperature difference.

Mechanical definition= wad-w

•If a quantity of heat is required to increase the temperature of a body by dT , the heat capacity C is defined to be

•Since heat is an inexact differential, depending upon the way in which energy changes occur, then it is clear that there cannot be a unique heat capacity for a system either.

dQC

dT

• So the quantity of heat which flows in will depend upon the path of the transformation and there will be an infinite number of heat capacities.

T

qC

Of the infinite possible number, it is customary to define two heat capacities. Heat capacity at constant volume CV and heat capacity at constant pressure Cp

dTCdq VV

CV is called heat capacity at constant volume.

Specific heat is essentially a measure of how thermally insensitive a substance is to the addition of energy.

CV can be determined by burning a known mass of substance that has known heat output. With CV known, it is simple to interpret an observed temperature rise as a release of heat

Most processes that occur in the laboratory, on the surface of the earth, and in organisms do so under a constant pressure of one atmosphere

Some of the energy supplied as heat to the system is returned to the surrounding as expansion work.

Specific heat is essentially a measure of how thermally insensitive a substance is to the addition of energy.

•For solids and liquids cV and cp are very similar.

•The high specific heat of water is the reason that coastal regions have milder climates than inland regions at the same latitude.

Aluminium 900 J Kg-1 C-1

Wood 1700 J Kg-1 C-1

Water 4186 J Kg-1 C-1

Important point about Internal energy

Internal energy is the total of the

kinetic energy of the constituent atoms or molecules due to their motion (translational, rotational, vibrational) and

the potential energy associated with intermolecular forces. It includes the energy in all of the chemical bonds, and the energy of the free, conduction electrons in metals.

Important point about Internal energy

• Internal energy does not include the translational or rotational kinetic energy of a body as a whole.

• It also does not include the relativistic mass-energy equivalent E = mc2.

• It excludes any potential energy a body may have because of its location in external gravitational or electrostatic field, although the potential energy it has in a field due to an induced electric or magnetic dipole moment does count.