# 19.5: The Temperature of a Gas Decreases in a Reversible Adiabatic Expansion

We can make the same argument for the heat along C. If we do the three processes A and B+C only to a tiny extent we can write:

And now we can integrate from V_{1} to V_{2} over the reversible adiabatic work along B and from T1 to T2 for the reversible isochoric heat along C. To separate the variables we do need to bring the temperature to the right side of the equation.:

The latter expression is valid for a reversible adiabatic expansion of a monatomic ideal gas (say Argon) because we used the C_{v} expression for such a system. We can use the gas law PV=nRT to translate this expression in one that relates pressure and volume see Eq 19.23