why specific heats at constant pressure and volume are the same for an incompressible substance?
The reason that they are very different fo things like ideal gases is that with a gas at constant volume, there is no PV work possible. All of the heat going into the gas goes to increasing the energy of the gas molecules. At constant pressure, as you heat the gas, the gas will expand against that pressure -- doing PV work. Some of the heat energy that you added to the gas went out of the gas into the surroundings as work. More heat needs to be supplied to get the same change in temperature.
For incompressible materials, is this possible? Can the substance do any PV work?