Why is specific heat capacity at constant pressure not equal to the rate of change of internal energy The formula for internal energy is $U= nC(v)dT$.
But why cant we use U=nC(p)dt where C(p) is the specific heat at constant pressure
Someone might say that we can use $C(v)$ because Internal energy of an ideal gas does not depend on the volume so even if volume is changing we can use $C(v)$.  But then the internal energy of an ideal gas is also not dependent on the pressure.  Its only depended on the temperature of the gas. So why cant we use $C(p)$?
Also for real gasses both $C(v)$, $C(p)$ shall be correct.  Isn't it? 
 A: The internal energy of a closed system is defined by:
$$ dU = \delta Q + \delta W $$
i.e. it is the energy added to the system plus the work done on the system. The work is $-PdV$ giving:
$$ dU = \delta Q - PdV $$
If we work at constant volume then $dV = 0$ and the internal energy is then simply the heat added i.e. $C_vdT$. If we work at constant pressure then the heat added is $C_pdT$, but we'd have to calculate the work done by the system as it expanded and then subtract that off to get the internal energy.
A: In Thermodynamics, the specific heat capacity at constant volume and the specific heat capacity at constant pressure are physical properties of a material (irrespective of process) that are precisely defined as follows:
$$C_v=\left(\frac{\partial U}{\partial T}\right)_V$$
and $$C_p=\left(\frac{\partial H}{\partial T}\right)_P$$where the specific internal energy U (per mole or per unit mass) and the specific enthalpy H (per mole or per unit mass) are also physical properties of the material (irrespective of process), with $$H=U+PV$$where V is the specific volume (per mole or per unit mass).  For an ideal gas, U and H are functions only of temperature, and not pressure or volume.  Therefore, for an ideal gas $$C_p=\left(\frac{dH}{dT}\right)=\frac{dU}{dT}+\frac{d(PV)}{dT}=C_v+\frac{d(RT)}{dT}=C_v+R$$
ADDENDUM
Problem 1:  This is a 2 step process involving 1 mole of ideal gas.  In Step 1, the system starts out at $P_1$, $V_1$, and $T_1$, and is heated at constant volume to temperature $T_2$ (and corresponding pressure $P_2$).  In Step 2, the gas is allowed to expand isothermally and reversibly (being held in contact with a constant temperature bath at $T_2)$ until the pressure is again $P_1$.
Step 1 Questions:
In terms of $T_1$, $T_2$, and $P_1$, what is the final pressure $P_2$ at the end of Step 1?
In terms of $T_1$ and $T_2$, what is $\Delta (PV)$ in Step 1?
In terms of $T_1$ and $T_2$, what is $\Delta U$, $\Delta H$, W, and Q in Step 1?
Step 2 Questions:
In terms of $T_1$, $T_2$, and $V_1$, what is the final volume at the end of Step 2?
What is $\Delta (PV)$ in Step 2?
In terms of $T_1$ and $T_2$, what is W and Q in Step 2?
What is $\Delta U$ and $\Delta H$ in Step 2?
Overall Process Questions:
In terms of $T_1$ and $T_2$, what are Q and W for the overall process?
In terms of $T_1$ and $T_2$, what are $\Delta U$ and $\Delta H$ for the overall process?
What is $\Delta U$ divided by $\Delta T = (T_2-T_1)$ for the overall process? 
What is $\Delta H$ divided by $\Delta T = (T_2-T_1)$ for the overall process?
