# 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. – Bob Jacobsen May 1 '18 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. – 8cold8hot May 1 '18 at 7:22
• What is the subscript sigma supposed to represent? – Chet Miller May 1 '18 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. – 8cold8hot May 1 '18 at 12:23
• There must be more to it than that. – Chet Miller May 1 '18 at 12:56

## 1 Answer

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.

• Thank you for the answer. Now I understand the deference between CsatL and CpL is the effect of pressure on entropy in determining the variation of liquid enthalpy with temperature along the saturation line. But how to link this effect to the statement that CsatL is the the energy required for a temperature change while maintaining the liquid in the saturated state. – 8cold8hot May 3 '18 at 8:30