What is the difference between the specific heat capacities of water under isobaric and isochoric conditions Can the difference of specific heat capacity of water under isochoric and isobaric conditions be explained in terms of the internal energy of the system? Most of the videos I have watched base their explanation in terms of ideal gases. I guess its something to do with the fact isochoric conditions mean all the heat energy provided goes to the internal energy of the molecules. I also have the graphs of the specific heat capacities plotted against time

 A: In general it is the same idea as with ideal gases. This here is not what is formal answer, because specific heat is generally defined with entalpy and internal energy. This is rather the explanation, why there is a difference.
In order to change volume $V$ when the pressure is constant, some work $A$ has to be provided. In differential case (very small change): $dA=pdV$.
From conservation of energy we can than determine that:
$$
dQ=dW+dA
$$
$Q$ is internal energy of a system, and dW is energy added, and A is work done by the system.
So we can denote specific heat as $dQ/dT$:
$$
c_p=\frac{dQ}{dT}=\frac{dW+pdV}{dT}
$$
and
$$
c_V=\frac{dQ}{dT}=\frac{dW+pdV}{dT}=\frac{dW+p\cdot0}{dT}=\frac{dW}{dT}
$$
You can see from here, that $c_p$ is greater than $c_v$. The relation between this two depends on equation of state and it can be quite ugly for liquids. But in general when we have isobaric conditions, some added energy is converted into work needed, to change the volume of system.
If you are not familiar with differentials $d$, they are just very small changes $\Delta$.
A: The specific heats diverge mainly after 100 C when at 1 atmosphere water changes phase and begins acting like a gas approaching  ideal gas behavior.
Based on the first law the internal energy explanation for the specific heat at constant pressure (isobaric) $C_P$ being greater than the specific heat at constant volume (isochoric) $C_V$ is because when heat is added at constant pressure the substance expands and does work. When added at constant volume it does no work. Based on the first law:
At constant pressure (isobaric):
$$Q=C_{P}\Delta T=\Delta U+W$$
At constant volume (isochoric) where $W=0$:
$$Q=C_{V}\Delta T=\Delta U+W=\Delta U$$
Assuming ideal gas behavior internal energy is a function of temperature only, from the above equations it takes more heat to achieve the same increase in internal energy (increase in temperature) in an isobaric process than an isochoric process because some of the heat does work. That requires $C_P$ to be greater than $C_V$.
Hope this helps
