Derive Stefan-Boltzmann's Law by Thermodynamic Method.
Derivation : Let's consider a black body that is in thermal equilibrium with its surroundings. According to the laws of thermodynamics, the entropy of a system in equilibrium is at a maximum. We can use this principle to derive Stefan-Boltzmann's law.
The entropy of a black body is given by the famous expression:
\(\displaystyle S = \frac{U}{T}\)
where \(S\) is the entropy, \(U\) is the total internal energy, and \(T\) is the absolute temperature. The internal energy of a black body can be expressed as the sum of its thermal energy and its radiant energy:
\(\displaystyle U = U_{\text{thermal}} + U_{\text{radiant}}\)
We can rewrite the entropy equation as:
\(\displaystyle S = \frac{U_{\text{thermal}} + U_{\text{radiant}}}{T}\)
Since the black body is in thermal equilibrium, its temperature is uniform throughout. Therefore, the temperature \(T\) is constant, and we can differentiate both sides of the entropy equation with respect to time:
\(\displaystyle \frac{dS}{dt} = \frac{d}{dt}\left(\frac{U_{\text{thermal}} + U_{\text{radiant}}}{T}\right)\)
Using the quotient rule, we can simplify the derivative on the right-hand side:
\(\displaystyle \frac{dS}{dt} = \frac{1}{T}\frac{d(U_{\text{thermal}} + U_{\text{radiant}})}{dt} - \frac{U_{\text{thermal}} + U_{\text{radiant}}}{T^2}\frac{dT}{dt}\)
Now, let's examine each term separately. The first term represents the rate of change of internal energy:
\(\displaystyle \frac{d(U_{\text{thermal}} + U_{\text{radiant}})}{dt} = \frac{dU_{\text{thermal}}}{dt} + \frac{dU_{\text{radiant}}}{dt}\)
The second term represents the rate of change of temperature:
\(\displaystyle \frac{dT}{dt}\)
Since the black body is in thermal equilibrium, the total internal energy and temperature do not change with time. Therefore, both the derivatives in the above equations are zero. Simplifying further, we obtain:
\(\displaystyle 0 = \frac{d(U_{\text{thermal}} + U_{\text{radiant}})}{dt} - \frac{U_{\text{thermal}} + U_{\text{radiant}}}{T}\frac{dT}{dt}\)
Rearranging the equation, we get:
\(\displaystyle \frac{d(U_{\text{thermal}} + U_{\text{radiant}})}{dt} = \frac{U_{\text{thermal}} + U_{\text{radiant}}}{T}\frac{dT}{dt}\)
Now, let's focus on the radiant energy term, \(U_{\text{radiant}}\). According to Planck's law, the energy radiated by a black body at temperature \(T\) is given by:
\(\displaystyle U_{\text{radiant}} = \sigma A T^4\)
where \(\sigma\) is the Stefan-Boltzmann constant and \(A\) is the surface area of the black body.
Substituting this expression for \(U_{\text{radiant}}\) into the equation, we have:
\(\displaystyle \frac{d(U_{\text{thermal}} + \sigma A T^4)}{dt} = \frac{(U_{\text{thermal}} + \sigma A T^4)}{T}\frac{dT}{dt}\)
Now, let's assume that the thermal energy \(U_{\text{thermal}}\) is independent of temperature, so its derivative with respect to temperature is zero:
\(\displaystyle \frac{dU_{\text{thermal}}}{dT} = 0\)
Therefore, the equation becomes:
\(\displaystyle \frac{d(\sigma A T^4)}{dt} = \frac{(\sigma A T^4)}{T}\frac{dT}{dt}\)
We can cancel out the \(A\) term from both sides of the equation:
\(\displaystyle \frac{d(\sigma T^4)}{dt} = \frac{(\sigma T^4)}{T}\frac{dT}{dt}\)
Now, let's integrate both sides of the equation with respect to time:
\(\displaystyle \int \frac{d(\sigma T^4)}{dt}\, dt = \int \frac{(\sigma T^4)}{T}\frac{dT}{dt}\, dt\)
The left-hand side can be integrated as:
\(\displaystyle \sigma T^4 = \int \frac{(\sigma T^4)}{T}\frac{dT}{dt}\, dt\)
Integrating the right-hand side yields:
\(\displaystyle \sigma T^4 = \int (\sigma T^3)\, dT\)
Simplifying further, we have:
\(\displaystyle \sigma T^4 = \frac{\sigma T^4}{4}\)
Finally, canceling out the \(\sigma\) and rearranging the equation, we arrive at the Stefan-Boltzmann law:
\(\displaystyle \sigma = \frac{1}{4}\)
Summary:
In this derivation, we started with the entropy equation and used the principles of thermodynamics to derive the Stefan-Boltzmann law. By assuming that the thermal energy is independent of temperature and incorporating Planck's law for black body radiation, we obtained the relationship between the radiant energy and the temperature of a black body. The derivation led us to the conclusion that the Stefan-Boltzmann constant, \(\sigma\), is equal to \(\frac{1}{4}\).
Also Read : What is the Planck's Quantum Hypothesis?
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