Why Use Cv in a Polytropic Process?

[Stonescar]

Homework Statement

A piston-cylinder device contains 1.2kg Nitrogen gas at 120 kPa and 300K. The gas is now compresses slowly in a polytropic process during which PV^1.3 = constant.
The process ends when the volume is reduced by one-half.
Determine the entropy change of Nitrogen during this process.

My question here is about what specific heat constant to use.

Homework Equations

I thought i had to use C_v when the volume is constant.
And C_p when the pressure was constant.

The Attempt at a Solution

As I can see, in this process. Neither the pressure or the volume is constant?
But in the solution provided, tey are using the C_v .
Why is that? The volume did get reduced by one-half

Im new to Thermodynamics, and not very good at it..
But this is confusing me. Can someone explain this to me? How to know when to use what constant :)

 

[Mapes]

Could the process have been divided into constant-volume and constant-pressure parts to simplify? This is a common strategy. But it's hard to say without seeing the solution.

Another possibility is that the constant-volume specific heat is being used in the equation for internal energy for an ideal gas,

$$E=mc_VT+E_0$$

which is always true (and can be used to find the change in internal energy of an ideal gas for any process if one knows the change in temperature).