Difference between revisions of "Thermodynamics of Black-Body Radiation"
| (2 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
[[Category:Cosmic Microwave Background (CMB)|1]] | [[Category:Cosmic Microwave Background (CMB)|1]] | ||
| − | + | __TOC__ | |
| − | + | ||
| − | + | ||
<div id="bbrazm1"></div> | <div id="bbrazm1"></div> | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 1 === | + | === Problem 1: chemical potential === |
Show that the photon gas in thermal equilibrium has zero chemical potential. | Show that the photon gas in thermal equilibrium has zero chemical potential. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 16: | Line 14: | ||
| − | + | In the next 4 problems we consider some volume $V$, filled with black-body radiation of temperature $T$. | |
<div id="bbrazm2"></div> | <div id="bbrazm2"></div> | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 2 === | + | === Problem 2: number density distribution === |
Find the number of photons with frequencies in the interval $[\omega ,\omega +d\omega]$. | Find the number of photons with frequencies in the interval $[\omega ,\omega +d\omega]$. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 28: | Line 26: | ||
</div> | </div> | ||
</div></div> | </div></div> | ||
| − | |||
| Line 34: | Line 31: | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 3 === | + | === Problem 3: total number === |
Find the total number of photons. | Find the total number of photons. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 42: | Line 39: | ||
</div> | </div> | ||
</div></div> | </div></div> | ||
| − | |||
| Line 48: | Line 44: | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 4 === | + | === Problem 4: gas oven === |
What is this number for a gas oven at room temperature and at maximum heat? | What is this number for a gas oven at room temperature and at maximum heat? | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 62: | Line 58: | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 5 === | + | === Problem 5: energy distribution === |
What is the energy of photons with frequencies in the interval $[ \omega ,\omega +d\omega]$? | What is the energy of photons with frequencies in the interval $[ \omega ,\omega +d\omega]$? | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 75: | Line 71: | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 6 === | + | === Problem 6: thermodynamic potentials === |
Calculate the free energy, entropy and total energy of black-body radiation. | Calculate the free energy, entropy and total energy of black-body radiation. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 87: | Line 83: | ||
<div id="bbrazm6"></div> | <div id="bbrazm6"></div> | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 7 === | + | === Problem 7: thermal capacity === |
Calculate the thermal capacity of black-body radiation. | Calculate the thermal capacity of black-body radiation. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 99: | Line 95: | ||
<div id="bbrazm7"></div> | <div id="bbrazm7"></div> | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 8 === | + | === Problem 8: pressure === |
Find the pressure of black-body radiation and construct its state equation. | Find the pressure of black-body radiation and construct its state equation. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 111: | Line 107: | ||
<div id="bbrazm8"></div> | <div id="bbrazm8"></div> | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 9 === | + | === Problem 9: adiabatic equation === |
Find the adiabatic equation for the black-body radiation. | Find the adiabatic equation for the black-body radiation. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 123: | Line 119: | ||
<div id="cmb3"></div> | <div id="cmb3"></div> | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 10 === | + | === Problem 10: CMB as microwave === |
Why CMB cannot be used to warm up food like in the microwave oven? | Why CMB cannot be used to warm up food like in the microwave oven? | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
| Line 136: | Line 132: | ||
<div id="cmb6"></div> | <div id="cmb6"></div> | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| − | === Problem 11 === | + | === Problem 11: Planck distribution === |
The binding energy of electron in the hydrogen atom equals to $13.6\ | The binding energy of electron in the hydrogen atom equals to $13.6\ | ||
eV$. What is the temperature of the Planck distribution with this | eV$. What is the temperature of the Planck distribution with this | ||
| Line 145: | Line 141: | ||
<p style="text-align: left;">The Planck distribution has maximum at frequency $\omega_m = 2.822 kT/\hbar.$ Then one obtains | <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> | $ kT = 13.6/2.822 = 4.82\mbox{eV}$ and $T\approx 5.6 \cdot {10^4}\mbox{K}$.</p> | ||
| + | </div> | ||
| + | </div></div> | ||
| + | |||
| + | |||
| + | |||
| + | <div id="cmb600"></div> | ||
| + | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| + | === Problem 12: power-law === | ||
| + | Find in power-law cosmology (see Chapter 3) time dependence of the CMB temperature $T(t)$. | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <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 | ||
| + | |||
| + | \[\frac{T_{0} }{T} =\frac{a}{a_{0} } =\left(\frac{t}{t_{0} } \right)^{\alpha } =\left(\frac{t}{t_{0} } \right)^{1/1+q} \] | ||
| + | |||
| + | where $q$is the deceleration parameter. Consequently, | ||
| + | |||
| + | \[T(t)=T_{0} \left(\frac{t}{t_{0} } \right)^{-\alpha } =T_{0} \left(\frac{t}{t_{0} } \right)^{-1/1+q} \]</p> | ||
</div> | </div> | ||
</div></div> | </div></div> | ||
Latest revision as of 23:18, 8 January 2013
Contents
- 1 Problem 1: chemical potential
- 2 Problem 2: number density distribution
- 3 Problem 3: total number
- 4 Problem 4: gas oven
- 5 Problem 5: energy distribution
- 6 Problem 6: thermodynamic potentials
- 7 Problem 7: thermal capacity
- 8 Problem 8: pressure
- 9 Problem 9: adiabatic equation
- 10 Problem 10: CMB as microwave
- 11 Problem 11: Planck distribution
- 12 Problem 12: power-law
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?
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 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?
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 $T\approx 5.6 \cdot {10^4}\mbox{K}$.
Problem 12: power-law
Find in power-law cosmology (see Chapter 3) time dependence of the CMB temperature $T(t)$.
In power-law cosmology the scale factor $a(t)$and the CMB temperature $T(t)$ are related through the relation \[\frac{T_{0} }{T} =\frac{a}{a_{0} } =\left(\frac{t}{t_{0} } \right)^{\alpha } =\left(\frac{t}{t_{0} } \right)^{1/1+q} \] where $q$is the deceleration parameter. Consequently, \[T(t)=T_{0} \left(\frac{t}{t_{0} } \right)^{-\alpha } =T_{0} \left(\frac{t}{t_{0} } \right)^{-1/1+q} \]
