Question: The adiabatic equation
In an adiabatic process no heat enters or leaves the system. It can be shown that if the state of a fixed quantity
of an ideal gas is changed adiabatically from pressure $P_1$ and volume $V_1$ to pressure $P_2$ and volume $V_2$ we have
that
$P_1V_1^{5/3} = P_2V_2^{5/3}$
i.e. during an adiabatic process
$PV^{5/3} = $ constant $= c_1$
This means that in an adiabatic change the temperature also changes. From the ideal gas law we have that
$PV/T = $ constant $= c_2$. Solving this for pressure we get $P = c_2T/V$, and substituting this in the adiabatic law
$(c_2T/V)V^{5/3} = c_1$; that is:
$TV^{2/3} = $ constant or $T_1V_1^{2/3} = T_2V_2^{2/3}$
So, when a gas is compressed adiabatically, the temperature increases. An everyday example of this is
pumping a bicycle tire: the valve gets very hot. This will be explained shortly.
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