divide both sides by : After integrating the left and right sides from to and from to and changing the sides respectively, Exponentiate both sides, substitute with , the heat capacity ratio and eliminate the negative sign to obtain Therefore, and At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (b4) gives Substituting this into (b2) gives Integrating we obtain the expression for work, Substituting in second term, Rearranging, Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases), By the continuous formula, or Substituting into the previous expression for , Substituting this expression and (b1) in (b3) gives Simplifying, 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 At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (c4) gives Substituting this into (c2) gives Integrating we obtain the expression for work, Substituting in second term, Rearranging, Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases), By the continuous formula, or Substituting into the previous expression for , Substituting this expression and (c1) in (c3) gives Simplifying, Graphing adiabats An adiabat is a curve of constant entropy in a diagram. Some properties of adiabats on a P–V diagram are indicated. These properties may be read from the classical behaviour of ideal gases, except in the region where PV becomes small (low temperature), where quantum effects become important. Every adiabat asymptotically approaches both the V axis and the P axis (just like isotherms). Each adiabat intersects each isotherm exactly once. An adiabat looks similar to an isotherm, except that during an expansion, an adiabat loses more pressure than an isotherm, so it has a steeper inclination (more vertical). If isotherms are concave towards the north-east direction (45°), then adiabats are concave towards the east north-east (31°). If adiabats and isotherms are graphed at regular intervals of entropy and temperature, respectively (like altitude on a contour map), then as