Thermal Physics Lecture Note 2

8/10/2019 Thermal Physics Lecture Note 2

1/23


2/23

From these graphs, it is observed that

3

0103143.8

PT

Pv J kmole

-1K

-1 for all temperatures and types of gas

The value R= 8.3143 103J kmole

-1K

-1 is called the Universal Gas Constant

is the Equation of State for Ideal GasnRTPVorRTPvorRT

=

Pv


3/23

2.2 Graphical representation of the equation of state :P-v-T surface

(i) With fixed T (ii) With fixed v

Pv = constantT

P= constant

(Boyle's Law)

(iii) WithP fixed,T

v= constant (Charles' Law)


4/23

3-D graph : P-v-T surface for ideal gas

v/T = constant

Pv = constant P/T = constant


5/23

2.3 Real gas

If the equation of state is not obeyed, that is

RT

Pv

the gas is said to be a real gas

The general form of the equation of state of a real gas is :

...........2

v

C

v

BAPv where A = RT

.........1 2RTv

C

RTv

BRTPv

The factor

.........12RTv

C

RTv

Bz is called the Departure Coefficient


6/23

If we take B = RTb a, C = RTb2 ............. and so on

v

a

v

bRT

v

a

v

bRT

v

bRTRT

v

RTb

v

aRTbRTPvHence

1

2

2

2

1

........

............,

where .........112

21

v

b

v

b

v

b

v

a

bv

vRTPv

2v

a

bv

RTP

Finally, RTbvvaP

=

2 - van der Waals equation


7/23

A gas that obeys this equation of state is called the van der Waals gas

In this model of "real gas", the effect of the intermolecular forces and the volume

occupied by the molecules themselves have been taken into consideration.

Intermolecular forces

P

2v

aP

Volume occupied by the molecules themselves v bv

Substance

a

(J m3kilomole-2)

b

(m3kilomole-1)

He 3.44 103 0.0234

H2 24.8 103 0.0266

O2 138 103 0.0318

CO2 366 103 0.0429

H2O 580 103 0.0319

Hg 292 103 0.0055


8/23

P-v curves P-v-T surfaces


9/23

2.4 Real substances :P-v-T surfaces

3 possible phases : solid, liquid, gas (vapor gas at equilibrium with its liquid)


10/23

(Reading : Section 2.5, pp 30-40)


11/23

2.5 Thermodynamic systems other thanP-v-T system (examples)

(a) A metal wire under tension

oo TTYA

LL 1 (Y: Young's modulus; : coeff. of linear expansion)

Thermodynamic properties: tension

,length L,

temperature T

(b) Paramagnetic material

T

HCM c - Curie's law (Cc: Curie constant)

Thermodynamic properties: magnetic field intensity H

magnetic moment M

temperature T


12/23

2.6 Partial derivatives

Process 1

2, 4

3 : isobaric

Process 1

4, 2

3 : isothermal

For isobaric process,

volume changes byvp

when temperature changes by Tp

whileP is not changed

Taking limit of Tp0 forp

p

T

v

, i.e.

pp

p

T T

v

T

v

p

=

0lim

- Partial derivative of vwith respect to T

pT

v

is the tangent at point 1


13/23

If vis given as a function ofPand T,

PT

v

can be obtained

Example : For ideal gas, Pv =RT

P

R

T

v

P

=

To calculate the change in the volume caused by the change in temperaturedTPduring

the isobaric process,

PP

P dT

T

vdv

=

The quantityPT

v

v

=

1 is the coefficient of volume expansion, or expansivity

For ideal gas,TP

RRTP

PR

v11

=

For finite temperature change, TP, the average value of expansivity is


14/23

P

P

T

v

v

=

1

1

Using the similar argument by referring to the v-P curve for the isothermal process,

Tangent at each point on the curve:TP

v

v

TT

T dPP

vdv

=

andTP

v

v

1 - isothermal compressibility

P

T

T

P

v1 - mean compressibility

v

1

when the pressure is increased isothermally by PT


15/23

In general, we can write for X=f(Y, Z) dZ

Z

XdY

Y

XdX

YZ

=

and hence, vdPvdTdPP

vdT

T

vdv

TP

=

or dPdTv

dv

Note that : expansivity is expressed in unit of ( K-1

)

and : compressibility is expressed in unit of ( m2N

-1)

The values of these quantities can be measured experimentally for a material.

Integrate the experssion fordv,

P

P

T

T

o

v

v ooo

vdPvdTvvdv

which gives ]oooooooo PPTTvPPvTTvvv 1


16/23

Example : copper


17/23

2.7 Critical constants of a van der Waals gas

For a van der Waals gas,2

v

a

bv

RTP

At the critical point, 0

Tv

P and 0

2

2

=

Tv

P


18/23

Hence, 02

)(32

=

cc

c

T v

a

bv

RT

v

P

and 06

)(

2

432

2

=

=

cc

c

Tv

a

bv

RT

v

P

Solving these equations gives : bvc 3 ,Rb

aTc

27

8= and

227b

aPc=

which are the system parameters at the critical point.

Hence by measuring these values experimentally, the values ofa andbfor the real gas can beobtained.

From the first equation,

3

cvb=

From the second and third equations,

c

c

P

RTb

8

=

This implies that at the critical point of a van der Waals gas, 375.08

3=

c

cc

RT

vP

Experimentally, this is found to be not true van der Waals gas model is not accurate!


19/23

2.8 Relations between partial derivatives

Volume differential : dPP

VdT

T

VdV

TP

=

Pressure differential : dVV

PdT

T

PdP

TV

=

Combine these two equations,

dTT

V

T

P

P

VdV

V

P

P

V

PVTTT

=

1

This equation should be obeyed by 2 nearby equilibrium states.

Consider the case when dT = 0 but dV 0, we obtain:

01 =

TT VP

PV , that is

T

T

V

PPV

1


20/23

Similarly, we can obtain for the case ofdT 0 whiledV= 0

0

PVT T

V

T

P

P

V

Hence 1

PVT V

T

T

P

P

V

In general, if f(X, Y, Z) = 0 , the we can write

Z

Z

XYY

X

1

1

YXZ X

Z

Z

Y

Y

X


21/23

2.9 Exact differentials

Taking state 1 : (P1V1T1)

state 2 : (P1V2T2)

state 3 : (P3V

3T

2)

state 4 : (P3V4T1)

For process 1 3, it can take 3 paths :

first path : 1

3

second path : 1 2 3

third path : 1 4 3

For path 1, it involves all three parameters P V T are varied together cannot be used !


22/23

Path 2, process 1 2 is isobaric (P = constant)

process 2 3 is isothermal ( T= constant)

Change in volume after these processes (volume differential) :

dPP

VdT

T

VdV

TP 21

321

=

Similarly, for path 3, dTT

VdP

P

VdV

PT 31

341

=

If we assume dV1-2-3=dV1-4-3 the volume differential is an exact differential

Then, we can write dTT

VdP

P

VdP

P

VdT

T

V

PTTP 3121

=

Hence,dT

P

V

P

V

dP

T

V

T

V

TTPP 1213

=


23/23

Take limit that dP 0 anddT 0 ,

TP

V

T

V

PdP

T

V

T

V

TP

PP

dP

=

=

2

0

13lim

PT

V

P

V

TdT

P

V

P

V

PT

TT

dT

=

=

2

0

12lim

This givesPT

V

TP

V

=

22

This means that the second partial derivatives of Vis independent of the order of

differentiation.

The differential of Vis then said to be an exact differential.

The differentials of all thermodynamic properties of a system are exact. In otherwords, a

quantity whose differential is not exact is not a thermodynamic property.

Thermal Physics Lecture Note 2

Documents

Transcript of Thermal Physics Lecture Note 2