Editing Adiabatic process (section)

===Derivation of discrete formula and work expression===
The change in internal energy of a system, measured from state 1 to state 2, is equal to

<!-- equation missing here? TODO: check revisions -->At the same time, the work done by the pressure–volume changes as a result from this process, is equal to

{{NumBlk||<math display="block"> W = \int_{V_1}^{V_2}P \,dV. </math>|{{EquationRef|c2}}}}

Since we require the process to be adiabatic, the following equation needs to be true

{{NumBlk||<math display="block"> \Delta U + W = 0. </math>|{{EquationRef|c3}}}}

By the previous derivation,

{{NumBlk||<math display="block"> P V^\gamma = \text{constant} = P_1 V_1^\gamma. </math>|{{EquationRef|c4}}}}

Rearranging (c4) gives

<math display="block"> P = P_1 \left(\frac{V_1}{V} \right)^\gamma. </math>

Substituting this into (c2) gives

<math display="block"> W = \int_{V_1}^{V_2} P_1 \left(\frac{V_1}{V} \right)^\gamma \,dV. </math>

Integrating we obtain the expression for work,

<math display="block"> W = P_1 V_1^\gamma \frac{V_2^{1-\gamma} - V_1^{1-\gamma}}{1 - \gamma} = \frac{P_2 V_2 - P_1 V_1}{1 - \gamma}. </math>

Substituting {{math|1=''γ'' = {{sfrac|''α'' + 1|''α''}}}} in second term,

<math display="block"> W = -\alpha P_1 V_1^\gamma \left( V_2^{1-\gamma} - V_1^{1-\gamma} \right). </math>

Rearranging,

<math display="block"> W = -\alpha P_1 V_1 \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right). </math>

Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases),

<math display="block"> W = -\alpha n R T_1 \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right). </math>

By the continuous formula,

<math display="block"> \frac{P_2}{P_1} = \left(\frac{V_2}{V_1}\right)^{-\gamma}, </math>

or

<math display="block"> \left(\frac{P_2}{P_1}\right)^{-\frac{1}{\gamma}} = \frac{V_2}{V_1}. </math>

Substituting into the previous expression for {{math|''W''}},

<math display="block"> W = -\alpha n R T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right). </math>

Substituting this expression and (c1) in (c3) gives

<math display="block"> \alpha n R (T_2 - T_1) = \alpha n R T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right). </math>

Simplifying,

<math display="block">\begin{align}
T_2 - T_1 &=  T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right), \\
\frac{T_2}{T_1} - 1 &=  \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1, \\
T_2 &= T_1 \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}}.
\end{align}</math>