Undergrad Relation between change of pressure and temperature in adiabatic process

[T C]

TL;DR

I am recently trying to formulate the relation between change in pressure and temperature for adiabatic process. Just submitting the math below for understanding of others. If there is any fault, then kindly rectify me.

In case of adiabatic process, we all know that the relation between temperature and pressure and that's given below:​

P. T^(γ/(1-γ)) = Const.
therefore, P = Const. T^{(γ/(γ - 1))}
or, ΔP = Const. (γ/(γ - 1)).ΔT^{(1/(γ - 1))}​

It's just an attempt to find out the relation. Don't know how much correct I am. Waiting for comments from others.​

 

[jack action]

No, it is:
$$P_0T_0^k = C$$
and:
$$PT^k = C$$
Therefore
$$P_0T_0^k = PT^k$$
Where $k = \frac{\lambda}{1-\lambda}$. Thus:
$$\Delta P = P- P_0$$
$$\Delta P= P_0\left(\left(\frac{T_0}{T}\right)^k-1\right)$$
$$\Delta P = P_0\left(\left(\frac{T}{T_0}\right)^{-k}-1\right)$$
$$\frac{\Delta P}{P_0} = \left(\frac{T_0 + \Delta T}{T_0}\right)^{-k}-1$$
Or:
$$1+ \frac{\Delta P}{P_0} = \left(1+\frac{\Delta T}{T_0}\right)^{\frac{\lambda}{\lambda - 1}}$$