My name is Alexander FufaeV and here I write about:
Adiabatic Process: How to Describe it with Adiabatic Equation
Important Formula
$$\class{red}{\mathit{\Pi}_1} ~=~ \class{blue}{\mathit{\Pi}_2} \, \left(\frac{\class{blue}{V_2}}{\class{red}{V_1}}\right)^{\gamma}$$
$$\class{red}{\mathit{\Pi}_1} ~=~ \class{blue}{\mathit{\Pi}_2} \, \left(\frac{\class{blue}{V_2}}{\class{red}{V_1}}\right)^{\gamma}$$
$$\class{red}{V_1} ~=~ \left( \frac{ \class{blue}{\mathit{\Pi}_2} }{ \class{red}{\mathit{\Pi}_1} } \, {\class{blue}{V_2}}^{\gamma} \right)^{-\gamma}$$
$$\gamma ~=~ \frac{ \ln(\class{red}{\mathit{\Pi}_1}) ~-~ \ln(\class{blue}{\mathit{\Pi}_2}) }{ \ln(\class{blue}{V_2}) ~-~ \ln(\class{red}{V_1})}$$
$$\class{blue}{\mathit{\Pi}_2} ~=~ \class{red}{\mathit{\Pi}_1} \, \left(\frac{\class{red}{V_1}}{\class{blue}{V_2}}\right)^{\gamma}$$
$$\class{blue}{V_2} ~=~ \left( \frac{ \class{red}{\mathit{\Pi}_1} }{ \class{blue}{\mathit{\Pi}_2} } \, {\class{red}{V_1}}^{\gamma} \right)^{-\gamma}$$
What do the formula symbols mean?
Pressure before
$$ \class{red}{\mathit{\Pi}_1} $$ Unit $$ \mathrm{Pa} $$
Pressure 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 is the quotient of heat capacities at constant pressure \( c_{\small{\Pi}} \) and volume \( c_{\small{\text V}} \).
For example, for monatomic gas: \( \gamma ~=~ \frac{5}{3} \).
Pressure after
$$ \class{blue}{\mathit{\Pi}_2} $$ Unit $$ \mathrm{Pa} $$
Pressure of the ideal gas AFTER the adiabatic process.
Volume after
$$ \class{blue}{V_2} $$ Unit $$ \mathrm{K} $$
Volume of the ideal gas AFTER the adiabatic process.
The adiabatic equation describes the relationship between the pressure \( \mathit{\Pi}\), volume \( V \) and temperature \( T \) of a gas during an adiabatic process. An adiabatic process is a thermodynamic process in which no heat is exchanged with the environment. This means that the change in the internal energy of the gas is exclusively due to the work performed on or by the gas.
$$ \begin{align} \class{red}{\mathit{\Pi}_1} ~=~ \class{blue}{\mathit{\Pi}_2} \, \left(\frac{\class{blue}{V_2}}{\class{red}{V_1}}\right)^{\gamma} \end{align} $$
$$ \begin{align} \gamma ~=~ \frac{ c_{\small{\Pi}} }{ c_{\small{\text V}} } \end{align} $$
$$ \begin{align} \class{red}{T_1} ~=~ \class{blue}{T_2} \, \left( \frac{\class{blue}{V_2}}{\class{red}{V_1}} \right)^{\gamma -1} \end{align} $$
$$ \begin{align} \class{red}{T_1} ~=~ \class{blue}{T_2} \, \left(\frac{\class{blue}{\mathit{\Pi}_2}}{ \class{red}{\mathit{\Pi}_1} }\right)^{ \frac{1}{\gamma}-1 } \end{align} $$