Formula Adiabatic Process Temperature before Pressure before Adiabatic index Temperature after Pressure after
$$\class{red}{T_1} ~=~ \class{blue}{T_2} \, \left(\frac{\class{blue}{\mathit{\Pi}_2}}{ \class{red}{\mathit{\Pi}_1} }\right)^{ \frac{1}{\gamma}-1 }$$ $$\class{red}{T_1} ~=~ \class{blue}{T_2} \, \left(\frac{\class{blue}{\mathit{\Pi}_2}}{ \class{red}{\mathit{\Pi}_1} }\right)^{ \frac{1}{\gamma}-1 }$$ $$\class{red}{\mathit{\Pi}_1} ~=~ \class{blue}{\mathit{\Pi}_2} \, \left( \frac{\class{blue}{T_2}}{\class{red}{T_1}} \right)^{1-\frac{1}{\gamma}}$$ $$\gamma ~=~ \left( \frac{ \ln(\class{red}{T_1}) ~-~ \ln(\class{blue}{T_2}) }{ \ln(\class{blue}{\mathit{\Pi}_2}) ~-~ \ln(\class{red}{\mathit{\Pi}_1}) } ~+~ 1 \right)^{-1}$$ $$\class{blue}{T_2} ~=~ \class{red}{T_1} \, \left(\frac{\class{red}{\mathit{\Pi}_1}}{ \class{blue}{\mathit{\Pi}_2} }\right)^{ \frac{1}{\gamma}-1 }$$ $$\class{blue}{\mathit{\Pi}_2} ~=~ \class{red}{\mathit{\Pi}_1} \, \left( \frac{\class{red}{T_1}}{\class{blue}{T_2}} \right)^{1-\frac{1}{\gamma}}$$
Temperature before
$$ \class{red}{T_1} $$ Unit $$ \mathrm{K} $$ Absolute temperature of the ideal gas before the adiabatic process.
Pressure before
$$ \class{red}{\mathit{\Pi}_1} $$ Unit $$ \mathrm{Pa} $$ Pressure of the ideal gas before the adiabatic process.
Adiabatic index
$$ \gamma $$ Unit $$ - $$ Adiabatic index is the quotient of heat capacities at constant pressure \( c_{\small{\Pi}} \) and volume \( c_{\small{\text V}} \).
For example, for monatomic gas: \( \gamma ~=~ \frac{5}{3} \).
Temperature after
$$ \class{blue}{T_2} $$ Unit $$ \mathrm{K} $$ Absolute temperature of the ideal gas after the adiabatic process.
Pressure after
$$ \class{blue}{\mathit{\Pi}_2} $$ Unit $$ \mathrm{Pa} $$ Pressure of the ideal gas AFTER the adiabatic process.