8. Applications of thermodynamics 8.1 Surface tension

The wire of length L is pulled to increase the area of the film A → A + dA Work done to the system dW = - σ dA σ : Surface tension (force/length)

We can take analogy to the system of liquid-vapour at equilibrium

When the volume is increased, V’’ ↓, V’’’ ↑ Liquid-surface at equilibrium The properties of the molecules at the surface are taken to be different from those in the bulk of the liquid When the surface area is increased, more molecules from the bulk of the liquid will be added to the surface

If the temperature is not changed during the process, σ will not be changed This implies that heat must be added to the system dAdQ λ= where λ : heat per unit surface area added, and

⎟⎠⎞

⎜⎝⎛−=

dTdT σλ

Consider isothermal process, dAdW σ−= dAdQ λ=

Internal energy : ( )dAdWdQdU σλ +=−= ⇒ ⎟⎠⎞

⎜⎝⎛−=+=⎟

⎠⎞

⎜⎝⎛∂∂

dTdT

AU

T

σσσλ

Lets say we start from A = 0 (U = 0)

∫∫ ⎟⎠⎞

⎜⎝⎛ −=

AU

dAdTdTdU

00

σσ

→ AdTdTU ⎟

⎠⎞

⎜⎝⎛ −=

σσ

Here U is the surface energy

dTdT

AU σσ −=

Define heat capacity for constant A

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−=⎟

⎠⎞

⎜⎝⎛∂∂

= 2

2

2

2

dTdAT

dTd

dTdT

dTdA

TUC

AA

σσσσ

Specific heat capacity : ⎟⎟⎠

⎞⎜⎜⎝

⎛−== 2

2

dTdT

ACc A

Aσ

Helmholtz function : VT

FTFU ⎟⎠⎞

⎜⎝⎛∂∂

−=

Compare AdTdTU ⎟

⎠⎞

⎜⎝⎛ −=

σσ ⇒ F = σ A

Entropy: ⎟⎠⎞

⎜⎝⎛−=⎟

⎠⎞

⎜⎝⎛∂∂

−=dTdA

TFS

A

σ [nota: V → A]

Specific entropy : dTds σ

−=

8.2 Vapour pressure of liquid drop A freely suspended liquid drop – assume spherical

σ (2πr)

(Pi - Pe) π r2

Consider one half of the drop, Net force : (Pi - Pe) π r2

Surface tension force : r)

σ (2π

(Pi - Pe) π r2 = σ (2πr)

∴ r

PP eiσ2

=−

⇒ r

PP eiσ2

+=

At thermodynamic equilibrium, Pe = P, the vapour pressure

∴ internal pressure r

PPiσ2

+=

This is the internal pressure of a liquid drop with radius r

8.3 Blackbody radiation

A blackbody is an object that absorp 100% radiation energy that is impinged onto it, and the radiation energy density emitted from it, u is expressed by Stefan Law

4Tu σ= (energy per unit volume) σ = 7.56 × 10-16 J m-3 K-4 Stefan constant Total internal energy : U = uV = σ T4 V

Heat capacitr for V unchanged : 34 VTTUC

VV σ=⎟

⎠⎞

⎜⎝⎛∂∂

=

Entropy : 3

0

2

0 3441 VTVTdTC

TS

T

V

T

σσ === ∫∫

Helmholtz function : 444

31

34 VTVTVTTSUF σσσ −=−=−=

Gibbs function : 031

31 44 =⎟

⎠⎞

⎜⎝⎛+−=+= VTVTPVFG σσ

8.4 Magnetic material Work : dW = - , dM where , is the external magnetic field strength M is the magnetic moment

When the magnetic material (paramagnetic) is placed into magnetic field, its potential energy is

Ep = - , M Hence, the total energy : E = U - , M [H = U + PV] dE = dU -, dM - Md, TdS equation : TdS = dU + dW = dE + , dM + Md, - , dM = dE + Md,

Compare with PVT system : TdS = dH – VdP In general, in our study of the thermodynamics of magnetism, we can take analogy of

PVT system :

Magnetic material PVT , P M VE H

C, CP

CM=MT

U⎟⎠⎞

⎜⎝⎛∂∂

VV T

UC ⎟⎠⎞

⎜⎝⎛∂∂

=

F = E - TS F = U - TS