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Stefan-Boltzman law

1. Stefan-Boltzman law

If U is the total energy density inside the box and u is the energy density per unit volume, and V the volume of the box; we can write: U = u V and dU = V du + u dV.

u is the energy density per volume unit and U is the (total) energy density of the blackbody.

If the box undergoes an isothermal variation of the volume, we can write:
dU = u dV.

du = du(T) = 0 (the energy inside the box depends only on the temperature.

The heat exchanged between the wall of the box and the vacuum inside the box that containing the radiations, can be written:

dQ = Cv dT + x dV, where Cv = V cv; and cv = dU/dT is the specific heat of the vacuum per volume unit.

For isothermal process, dQ = x dV.

The work exchanged between the wall of the box and the vacuum inside is dW = - pdV, where p = u/3, the related radiation pressure per volume unit.

The thermodynamic first law can be written as:

For the system ≡ vacuum inside the box, dQ = dU - dW or dU = dQ + dW = x dV - (u/3) dV. Therfore:

dQ = u dV = x dV - (u/3) dV → x = u + u/3 = 4u/3.
dQ becomes;

dQ = V (dU/dT)dT + (4u/3)dV

We assume that the process is reversible, the variation of the entropy is:
dS = dQ/T = (V/T)(du/dT)dT + (4/3)(u/T) dV.
The second thermodynamic principle expresses the dS is a total differential, that is:
∂/∂V [(V/T)(du/dT)]T = constant = ∂/∂T [(4/3)(u/T)]V = constant
We get:

[(1/T)(du/dT)] = (4/3)[T (du/dT) - u]/T2
du/dT = (4/3)[T (du/dT) - u]/T = (4/3)[ (du/dT) - u/T]
→ du/dT - (4/3)(du/dT) = - (4/3)(u/T)
(- 1 /3)(du/dT) = - (4/3)(u/T) → (du/dT) = (4)(u/T) → du/u = 4 (dT/T)
→ dU/U = 4 (dT/T) → ln (U)= 4 ln (T) → U = a T4

The constant a is the radiation constant.

U(T) = aT4

Stefan law

The energy density of the blackbody's radiation is proportional to the fourth power of the temperature. U(T) is a universal temperature function.

2. Stefan's Constant:

We have seen that the emissive power of the blackbody is P = π E(T). With the expression of the total density density U(T) = (4π/c) E(T) and the expression of the total energy, we get : P = (π U(T)/c)/4&pi = (ac/4)T4.
P = σT4

The constant σ = ac/4 is called the Stefan constant; a is the radiation constant. The Stefan Law and its constant were established by the physicist Stefan in 1879 empirically. Later The physicist Boltzman gave to the Stefan's formula a thermodinamic derivation.

The value of the Stefan's constant σ was found experimentally.
σ = 5.670 x 10 -8 W m-2 K-4.

3. Related thermodynamic expressions of the blackbody

1.1. The total energy:

U(T) = a T4
Where the radiation constant a = 4σ/c.
The value of the speed of light is 2.998 x 108 m/s (c = 299,792,458 metres per second).
a = 4σ/c = 7.5657 x 10-16 J m-3 K-4

1.2. The specific heat in the vacuum of the cavity:

cv = dU/dT - 4 a T3 That is the heat quatity to provide in order to increase by 1 C0 the temperature of the radiation in the cavity of the blackbody.

1.3. The specific entropy in the blackbody:

Since the entropy dS = dQ/T = cv dT/T = 4 a T2 dT, and according to the NernsT 's principle (S = 0 at T = 0); the integration gives:
S = (4/3) a T3

1.4. The specific free energy:

F = U - TS = a T4 - (4/3) a T4 = - (a/3) T4
F = - (1/3) a T4

1.5. The pressure of the radiation in the cavity:

P = U(T)/3 = (1/3) a T4

1.6. The free enthalpy the cavity:

G = F + PV = (- 1/3) a T4 + (- 1/3) a T4 = 0
The free enthaply of the blackbody is null.

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