Leis da termodinâmica

Algumas manipulações:

[latex]\\\mathrm{P_A\times V_A^{\gamma }=P_B\times V_B^{\gamma }\to P_0\times V_A^{\gamma }=r\times P_0\times V_B^{\gamma }\to V_A= r^{\frac{1}{\gamma }}\times V_B}\\\\\mathrm{P_D\times V_D^{\gamma }=P_C\times V_C^{\gamma }\to P_0\times V_D^{\gamma }=r\times P_0\times V_C^{\gamma }\to V_D= r^{\frac{1}{\gamma }}\times V_C}\\\\\mathrm{\tau _{B\to C}=P_B\times (V_C-V_B)=r\times P_0\times (V_C-V_B)}\\\\\mathrm{\tau _{D\to A}=P_D\times (V_A-V_D)=-r^{\frac{1}{\gamma }}\times P_0\times (V_C-V_B)}\\\\\mathrm{\tau _{A\to B}=\frac{V_A\times P_A-V_B\times P_B}{\gamma -1}=-P_0\times V_B\times \frac{r-r^{\frac{1}{\gamma }}}{\gamma -1}}\\\\\mathrm{\tau _{C\to D}=\frac{V_C\times P_C-V_D\times P_D}{\gamma -1}=P_0\times V_C\times \frac{r-r^{\frac{1}{\gamma }}}{\gamma -1}}\\\\\mathrm{dU_{B\to C}=n\times c_v\times dT=n\times \frac{R}{\gamma -1}\times T_C-n\times \frac{R}{\gamma -1}\times T_B}\\\\\mathrm{dU_{B\to C}=\frac{1}{\gamma -1}(r\times P_0\times V_C-r\times P_0\times V_B)=\frac{r\times P_0\times (V_C-V_B)}{\gamma -1}}[/latex]

Da relação que nos fornece o rendimento e mais algumas manipulações:

[latex]\\\mathrm{\eta =\frac{\tau }{Q}=\frac{\tau _{A\to B}+\tau _{B\to C}+\tau _{C\to D}+\tau _{D\to A}}{\tau _{B\to C}+dU_{B\to C}}=\underline{1-\left ( \frac{1}{r} \right )^{\frac{\gamma -1}{\gamma }}}}[/latex]

Acho que é isso.