Difference between revisions of "Thermodynamics of Black-Body Radiation"

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[[Category:Cosmic Microwave Background (CMB)|1]]
 
[[Category:Cosmic Microwave Background (CMB)|1]]
=Thermodynamics of Black-Body Radiation=
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__TOC__
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<div id="bbrazm1"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1: chemical potential ===
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Show that the photon gas in thermal equilibrium has zero chemical potential.
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  <div class="NavHead">solution</div>
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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In the next 4 problems we consider some volume $V$, filled with black-body radiation of temperature $T$.
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<div id="bbrazm2"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 2: number density distribution ===
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Find the number of photons with frequencies in the interval $[\omega ,\omega +d\omega]$.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="bbrazm3"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 3: total number ===
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Find the total number of photons.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="cmb_td_1"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 4: gas oven ===
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What is this number for a gas oven at room temperature and at maximum heat?
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="bbrazm4"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 5: energy distribution ===
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What is the energy of photons with frequencies in the interval $[ \omega ,\omega +d\omega]$?
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="bbrazm5"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 6: thermodynamic potentials ===
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Calculate the free energy, entropy and total energy of black-body radiation.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="bbrazm6"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 7: thermal capacity ===
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Calculate the thermal capacity of black-body radiation.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="bbrazm7"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 8: pressure ===
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Find the pressure of black-body radiation and construct its state equation.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 9: adiabatic equation ===
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Find the adiabatic equation for the black-body radiation.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="cmb3"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 10: CMB as microwave ===
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Why CMB cannot be used to warm up food like in the microwave oven?
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">The relic radiation, or CMB, corresponds to the black-body radiation with temperature $T_{CMB}=2.725\:K.$
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According to the main principle of thermodynamics, heat cannot transfer from a less heated body to more heated one, and thus the body (food in our case), which initially had temperature $T_0>T_{CMB},$ will emit more energy in the environment then absorb back, until the equilibrium installs with the CMB radiation at temperature $T_{CMB}.$</p>
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  </div>
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</div></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 11: Planck distribution ===
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The binding energy of electron in the hydrogen atom equals to $13.6\
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eV$. What is the temperature of the Planck distribution with this
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average photon energy?
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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    <p style="text-align: left;">The Planck distribution has maximum at frequency $\omega_m = 2.822 kT/\hbar.$ Then one obtains
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$ kT = 13.6/2.822 = 4.82\mbox{eV}$ and $T\approx 5.6 \cdot {10^4}\mbox{K}$.</p>
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  </div>
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</div></div>
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<div id="cmb600"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 12: power-law ===
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Find in power-law cosmology (see Chapter 3) time dependence of the CMB temperature $T(t)$.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">In power-law cosmology the scale factor $a(t)$and the CMB temperature $T(t)$ are related through the relation
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\[\frac{T_{0} }{T} =\frac{a}{a_{0} } =\left(\frac{t}{t_{0} } \right)^{\alpha } =\left(\frac{t}{t_{0} } \right)^{1/1+q} \]
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where $q$is the deceleration parameter. Consequently,
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\[T(t)=T_{0} \left(\frac{t}{t_{0} } \right)^{-\alpha } =T_{0} \left(\frac{t}{t_{0} } \right)^{-1/1+q} \]</p>
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  </div>
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</div></div>

Latest revision as of 23:18, 8 January 2013

Problem 1: chemical potential

Show that the photon gas in thermal equilibrium has zero chemical potential.


In the next 4 problems we consider some volume $V$, filled with black-body radiation of temperature $T$.

Problem 2: number density distribution

Find the number of photons with frequencies in the interval $[\omega ,\omega +d\omega]$.


Problem 3: total number

Find the total number of photons.


Problem 4: gas oven

What is this number for a gas oven at room temperature and at maximum heat?


Problem 5: energy distribution

What is the energy of photons with frequencies in the interval $[ \omega ,\omega +d\omega]$?


Problem 6: thermodynamic potentials

Calculate the free energy, entropy and total energy of black-body radiation.


Problem 7: thermal capacity

Calculate the thermal capacity of black-body radiation.


Problem 8: pressure

Find the pressure of black-body radiation and construct its state equation.


Problem 9: adiabatic equation

Find the adiabatic equation for the black-body radiation.


Problem 10: CMB as microwave

Why CMB cannot be used to warm up food like in the microwave oven?


Problem 11: Planck distribution

The binding energy of electron in the hydrogen atom equals to $13.6\ eV$. What is the temperature of the Planck distribution with this average photon energy?


Problem 12: power-law

Find in power-law cosmology (see Chapter 3) time dependence of the CMB temperature $T(t)$.