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]] | ||
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+ | |||
+ | __NOTOC__ | ||
+ | |||
+ | <div id="bbrazm1"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Show that the photon gas in thermal equilibrium has zero chemical potential. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="bbrazm2"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Find number of photons in black-body radiation of temperature $T$ in volume $V$, which have frequencies in the interval $\left[ \omega ,\omega +d\omega \right]$. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="bbrazm3"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Find total photon number of black body radiation in volume $V$ at temperature $T$. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="cmb_td_1"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Estimate number of photons in a gas oven at room temperature and at maximum heat. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="bbrazm4"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Find energy of photons in black-body radiation of temperature $T$ in volume $V$, which have frequencies in the interval $\left[ \omega ,\omega +d\omega \right]$. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="bbrazm5"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Calculate free energy, entropy and total energy of black-body radiation. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="bbrazm6"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Calculate thermal capacity of black-body radiation. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="bbrazm7"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Find pressure of black-body radiation and construct its state equation. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="bbrazm8"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Find adiabate equation for the photon gas of black-body radiation. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="cmb3"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Why CMB cannot be used to warm up food like in the microwave oven? | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">The relic radiation, or CMB, corresponds to the black-body radiation with temperature $T_{CMB}=2.725\:K.$ | ||
+ | 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> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="cmb6"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | The binding energy of electron in the hydrogen atom equals to $13.6\ | ||
+ | eV$. What is the temperature of Planck distribution, with this | ||
+ | average photon energy? | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">The Planck distribution has maximum at frequency $\omega_m = 2.822 kT/\hbar.$ Then one obtains | ||
+ | $ kT = 13.6/2.822 = 4.82\mbox{eV}$ and $T\approx 5.6 \cdot {10^4}\mbox{K}$.</p> | ||
+ | </div> | ||
+ | </div></div> |
Revision as of 19:26, 30 September 2012
Problem 1
Show that the photon gas in thermal equilibrium has zero chemical potential.
Problem 1
Find number of photons in black-body radiation of temperature $T$ in volume $V$, which have frequencies in the interval $\left[ \omega ,\omega +d\omega \right]$.
Problem 1
Find total photon number of black body radiation in volume $V$ at temperature $T$.
Problem 1
Estimate number of photons in a gas oven at room temperature and at maximum heat.
Problem 1
Find energy of photons in black-body radiation of temperature $T$ in volume $V$, which have frequencies in the interval $\left[ \omega ,\omega +d\omega \right]$.
Problem 1
Calculate free energy, entropy and total energy of black-body radiation.
Problem 1
Calculate thermal capacity of black-body radiation.
Problem 1
Find pressure of black-body radiation and construct its state equation.
Problem 1
Find adiabate equation for the photon gas of black-body radiation.
Problem 1
Why CMB cannot be used to warm up food like in the microwave oven?
The relic radiation, or CMB, corresponds to the black-body radiation with temperature $T_{CMB}=2.725\:K.$ 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 $T_{CMB}.$
Problem 1
The binding energy of electron in the hydrogen atom equals to $13.6\ eV$. What is the temperature of Planck distribution, with this average photon energy?
The Planck distribution has maximum at frequency $\omega_m = 2.822 kT/\hbar.$ Then one obtains $ kT = 13.6/2.822 = 4.82\mbox{eV}$ and $T\approx 5.6 \cdot {10^4}\mbox{K}$.