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 Problem Statement
 Show that ##pV^\gamma## is a constant for an adiabatic process
 Relevant Equations

Assume that gases behave according to a law given by ##pV = f(T)##, where ##f(T)## is a function of temperature. I have derived the following results:
##\displaystyle\left(\frac{\partial p}{\partial T}\right)_V=\frac{1}{V}\frac{\partial f}{\partial T}##  (1)
##\displaystyle\left(\frac{\partial V}{\partial T}\right)_p=\frac{1}{p}\frac{\partial f}{\partial T}##  (2)
##\displaystyle\left(\frac{\partial Q}{\partial V}\right)_p=C_p\left(\frac{\partial T}{\partial V}\right)_p##  (3)
##\displaystyle\left(\frac{\partial Q}{\partial p}\right)_V=C_V\left(\frac{\partial T}{\partial p}\right)_V##  (4)
Now,
## \displaystyle dQ=\left(\frac{\partial Q}{\partial p}\right)_V dp+\left(\frac{\partial Q}{\partial V}\right)_p dV##
In an adiabatic change, ##dQ=0##.
So, ## \displaystyle\left(\frac{\partial Q}{\partial p}\right)_V dp+\left(\frac{\partial Q}{\partial V}\right)_p dV=0##
Using (3) and (4),
##\displaystyle C_V\left(\frac{\partial T}{\partial p}\right)_V dp+C_p\left(\frac{\partial T}{\partial V}\right)_p dV=0##
Dividing this equation by ##C_V##, we get
##\displaystyle\left(\frac{\partial T}{\partial p}\right)_V dp+\gamma \left(\frac{\partial T}{\partial V}\right)_p dV=0##
How do I proceed?
Note: I know there may be lots of ways (some easier than this) of showing that ##pV^\gamma## is a constant for an adiabatic process. But this is the method required by my textbook.
## \displaystyle dQ=\left(\frac{\partial Q}{\partial p}\right)_V dp+\left(\frac{\partial Q}{\partial V}\right)_p dV##
In an adiabatic change, ##dQ=0##.
So, ## \displaystyle\left(\frac{\partial Q}{\partial p}\right)_V dp+\left(\frac{\partial Q}{\partial V}\right)_p dV=0##
Using (3) and (4),
##\displaystyle C_V\left(\frac{\partial T}{\partial p}\right)_V dp+C_p\left(\frac{\partial T}{\partial V}\right)_p dV=0##
Dividing this equation by ##C_V##, we get
##\displaystyle\left(\frac{\partial T}{\partial p}\right)_V dp+\gamma \left(\frac{\partial T}{\partial V}\right)_p dV=0##
How do I proceed?
Note: I know there may be lots of ways (some easier than this) of showing that ##pV^\gamma## is a constant for an adiabatic process. But this is the method required by my textbook.
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