Specific heat capacity for Adiabatic process

Question

We can define infinite number of specific heat capacities between two temperatures(e.g $c_p ,c_v$) . Could we do that for an adiabatic process. As for adiabatic process heat addition or removal is zero.

I asked this question because I found the work input equation for joule cycle defined for gas turbine unit and the equation is $$ c_p(T_2 - T_1)$$

How it is derived from STEADY FLOW ENERGY EQUATION. Work input is the work of compressor. Similarly we have turbine output equation as $$c_p(T_3 -T_4)$$

Where 1 to 2 is isentropic compression and 3 to 4 is isentropic expansion.

Answer 1

Specific heat capacity is defined for a substance not for a process. Specific heat capacity depends on the structure of the substance but, it ($C_P$ for example) can be measured by the formula $$C_P=\left(\large{\frac {\partial h}{\partial T}}\right)_P=\left(\large{\frac {\partial q}{\partial T}}\right)_P$$

$C_P$ isn't a function of heat ($q$) or kind of process (isobaric, isochoric, isothermal, etc.)

As an obvious example, we can measure the volume of a container by measuring the volume of water that can fill it. But, all of us know that volume of a container depends on its dimensions not on volume of the water (If we don't have any water, doesn't container have a quantity called volume?!!:-)