Water anomaly -- Heat capacities

[LagrangeEuler]

From relation
C_p-C_v=[P+(\frac{\partial U}{\partial V})_T](\frac{\partial V}{\partial T})_P
In case of water $(\frac{\partial V}{\partial T})_P<0$ so $C_p<C_v$? Right?

 

[ScifiGen]

From relation
C_p-C_v=[P+(\frac{\partial U}{\partial V})_T](\frac{\partial V}{\partial T})_P
In case of water $(\frac{\partial V}{\partial T})_P<0$ so $C_p<C_v$? Right?

(∂V/∂T)P is the amount of work done.
If we consider this amount of work done, in Cp, i.e, at constant pressure heat is absorbed and work is done, not only to increase the temperature of the gas (or the Substance) but also to increase the volume of the gas (Which, isn't constant here) which includes increasing both potential energies of the molecules and doing work against outside pressure, if any.For Cv, the whole of the energy spent and work done in terms of increase in volume is absent as this is done at constant volume. thus less work in done here. Hence, for gases, Cp>Cv.
P.S:
Mayer's Relation
Mayer derived a relation between specific heat at constant pressure and the specific heat at constant volume for an ideal gas. The relation is:

,

where CP,m is the specific heat at constant pressure, CV,m is the specific heat at constant volume and R is the gas constant.

For more general homogeneous substances( which can be water), not just ideal gases, the difference takes the form,

(see relations between heat capacities), where

is the heat capacity of a body at constant pressure,

is the heat capacity at constant volume,

is the volume,

is the temperature,

is the thermal expansion coefficient and

is the isothermal compressibility.

From this relation, several inferences can be made:

• Since isothermal compressibility
  
  is positive for all phases and the square of thermal expansion coefficient
  
  is a positive quantity or zero, the specific heat at constant-pressure is always greater than or equal to specific heat at constant-volume.

≥