3
$\begingroup$

I am making a comparation between the photon gas and the ideal classic gas for my Thermodynamics class. The photon gas is defined by the equations:

$$U=aVT^4 $$ $$P=\dfrac{1}{3}aT^4$$

I found this document: http://www.csupomona.edu/~hsleff/PhotonGasAJP.pdf which explain how to find some basic things, like enthalpy and entropy. It says that a great exercise is to compare the Carnot cycle of the photon gas with the Carnot cycle of the ideal gas. According to it, the efficiency is $\eta=1-\frac{T_2}{T_1}$ the same as the ideal gas. I think that this is really interesting for my comparation, so I'm trying to calculate the Carnot cycle efficiency for this gas.

I have no problem with the isothermal process, which is solved in that document: $$W_{ab}=-\dfrac{1}{3}aT^4\Delta V$$

However, I'm not sure if my result of the adiabatic process is correct. Work is $W=\int PdV$. Now, I can use the photon gas adiabatic equation (see the document) $PV^{4/3}=k$, where $k$ is a constant, to substitute $P$ in work equation, and integrate to obtain:

$$W_{bc} =\dfrac{3}{4}k\left( \dfrac{1}{V_b^3} - \dfrac{1}{V_c^3} \right)$$

I'm not sure if this result is correct. When I try to calculate the efficiency of the cycle, I have: $$\eta=\dfrac{|W_T|}{|Q_{ab}|}=\dfrac{|W_{ab}+W_{bc}+W_{cd}+W_{da}|}{|Q_{ab}|}$$

where $W_{ab}$,$W_{cd}$ are isothermal and $W_{bc}$,$W_{da}$ are adiabatic. The heat is also defined in the document as:

$$Q_{ab}=\dfrac{4}{3}aT^4\Delta V$$

But with these values I can't obtain the correct expression for the efficiency, or I don't know how to reduce the efficiency expression to obtain what I want. Which is the correct way to calculate adiabatic work in a photon gas? And the Carnot cycle efficiency?

Thank you all for your answers :D

$\endgroup$
1
$\begingroup$

I finally managed to solve the problem. Using that $\Delta U _ {cycle} = 0$, knowing the heats $Q_{ab}$ and $Q_{cd}$ and the adiabatic equation $PT^3=constant$. is possible to use the first law to find the total work expression in the cycle, and substitute in the efficiency expression.

You can find the problem solved step-by-step here: http://folk.uio.no/yurig/fys203/oppgaver/reichl.tmp.pdf I finally found this on Google, after a lot of time.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.