Why is specific heat at constant pressure greater than specific heat at constant volume?

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Basic Explanation: 

 When a gas is heated at constant pressure, the heat supplied to the gas is utilized in the following ways. 
  1. To raise the temperature of the gas. The heat remains within the body and represents the increase in internal energy. dU = m Cv (T2-T1) 
  2. To do some external work during expansion. W = p (V2-V1) = m R (T2-T1)
Thus, specific heat at constant pressure is greater the specific heat at constant volume.

Mathematical Explanation: 

 According to the first law of thermodynamics Q = ΔU + W Where, (W=PΔV)

Internal energy measures the temperature of the body. 

At constant volume as there is no work done on the gas so ΔV = 0. All the heat supplied utilize an increase in internal energy. Hence, the temperature of the body increases with less supply of heat. Whereas, at constant pressure, the supply of heat accounts for both internal energy and gas work. Thus, more heat is required to increase the temperature. 

Thus, Cp > Cv

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