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 😀
$$W_{total}=nR\left ( T_h\ln \frac{V_2}{V_1}-T_c\ln\frac{V_3}{V_4} \right )$$
Also for adiabatic processes we have :
$$T_h V_2^{\gamma-1 }=T_c V_3^{\gamma-1}\tag{2}$$
$$T_hV_1^{\gamma-1}=T_cV_4^{\gamma-1}\tag{3}$$
Substituting for $V_2$ and $V_4$ from $(2)$ and $(3)$ in the first equation, we can find the ratio $V_3\over V_1$. (Find the final result yourself!)
Best Answer
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.