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Cp is higher than Cv, because when pressure is constant, volume is not, so the gas is doing work via expansion. Therefore, you must account for the fact that for Cp, not all of the heat will be used toward increasing the temperature of the substance; some will be lost to work. To answer why pressure and volume can't both be constant, the ideal gas equation, PV = nRT, is useful. If heat is added, then T will increase. If we momentarily assume that P and V do not change, then that must mean that n and/or R must change. n will not change b/c we are not adding any more moles of substance when adding heat. R can also not change because it is a constant. Thus, pressure and volume can't both be constant: one of them has to change.
CV is the specific heat at constant volume, and CP is the specific heat at constant pressure. Specific heat is the heat energy required to raise the temperature of a substance (per unit mass) by one degree Celsius. The main difference between CV and CP is that the volume change is zero for a system under CV whereas the pressure change is zero for a system under CP.
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