Thermal Physics Lecture Note 8
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Transcript of Thermal Physics Lecture Note 8
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,