Formula Adiabatic Process Temperature before Volume before Adiabatic index Temperature after Volume after
$$\class{red}{T_1} ~=~ \class{blue}{T_2} \, \left( \frac{\class{blue}{V_2}}{\class{red}{V_1}} \right)^{\gamma -1}$$ $$\class{red}{T_1} ~=~ \class{blue}{T_2} \, \left( \frac{\class{blue}{V_2}}{\class{red}{V_1}} \right)^{\gamma -1}$$ $$\class{red}{V_1} ~=~ \class{blue}{V_2} \, \left( \frac{\class{blue}{T_2}}{\class{red}{T_1}} \right)^{\gamma - 1}$$ $$\gamma ~=~ \frac{ \ln(\class{red}{T_1}) ~-~ \ln(\class{blue}{T_2}) }{ \ln(\class{blue}{V_2}) ~-~ \ln(\class{red}{V_1}) } ~+~ 1$$ $$\class{blue}{T_2} ~=~ \class{red}{T_1} \, \left( \frac{\class{blue}{V_2}}{\class{red}{V_1}} \right)^{\gamma -1}$$ $$\class{blue}{V_2} ~=~ \class{red}{V_1} \, \left( \frac{\class{red}{T_1}}{\class{blue}{T_2}} \right)^{\gamma - 1}$$
Temperature before
$$ \class{red}{T_1} $$ Unit $$ \mathrm{K} $$ Absolute temperature of the ideal gas BEFORE the adiabatic process.
Volume before
$$ \class{red}{V_1} $$ Unit $$ \mathrm{m}^3 $$ Volume of the ideal gas BEFORE the adiabatic process.
Adiabatic index
$$ \gamma $$ Unit $$ - $$ Adiabatic index (also called heat capacity ratio) is the quotient of thermal 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.
Volume after
$$ \class{blue}{V_2} $$ Unit $$ \mathrm{m}^3 $$ Volume of the ideal gas AFTER the adiabatic process.