# What is the difference between these three definitions of specific liquid heat capacity? This is an excerpt from the page 6.18 of book "Properties of gases and liquids, 5th ed". I can figure out the difference between the first one C_pL with other two, but cannot distinguish latter two, C_sigmaL and C_satL. So I fail to derive the rightmost expression in equation (6-6.1). I also looked for other references [1-2] talking about specific liquid heat capacities near the critical point. But none of them makes distinguish among these three definitions of specific liquid heat capacities.

Ref: 1)Hall, W. B. "Heat transfer near the critical point." In Advances in Heat Transfer, vol. 7, pp. 1-86. Elsevier, 1971. 2)Hendricks, Robert C., Robert J. Simoneau, and Ray V. Smith. "Survey of heat transfer to near-critical fluids." In Advances in Cryogenic Engineering, pp. 197-237. Springer, Boston, MA, 1995.

• Not entirely sure what you're asking. Cp is the change in enthalpy with temperature at constant pressure; Cσ is the change in enthalpy with temperature for the saturated liquid case; Csat is the energy required for a temperature change while maintaining the liquid in the saturated state. May 1, 2018 at 5:37
• It seems to me that Csat and Cσ are the same. I suppose the energy for a temperature change while maintaining the liquid in the saturated state is the enthalpy change needed for raising the temperature of the saturated liquid. May 1, 2018 at 7:22
• What is the subscript sigma supposed to represent? May 1, 2018 at 12:17
• The book does not mention the meaning of subscript sigma in the text or in the notation list. I suppose it is just used for distinguishing it from others. May 1, 2018 at 12:23
• There must be more to it than that. May 1, 2018 at 12:56

It looks like what they are calling $C_{satL}$ is really the following: $$C_{satL}=C_{pL}-\left[T\left(\frac{\partial V}{\partial T}\right)_P\right]\left(\frac{dP}{dT}\right)_{\sigma L}=C_{pL}+T\left(\frac{\partial S}{\partial P}\right)_T\left(\frac{dP}{dT}\right)_{\sigma L}$$The second term in this equation takes into account the effect of pressure on entropy in determining the variation of liquid enthalpy with temperature along the saturation line.