RMP Lecture Notes

Specific Heat

In this class, we will generally use the specific heat or heat capacity whenever thermodynamic calculations are made using the ideal gas model. We can also use them for other substances when we lack good data tables.

The specific heat is the amount of heat needed to raise the temperature of a unit mass of substance by one degree:

Heat is a path dependent property, so the calculation requires us to specify the path used. This leads to two different values of the specific heat:

c_v to calculate the heat required via a constant volume path
c_p to calculate the heat required via a constant pressure path

Calculating the heat is useful, but it would be useful to see how specific heat relates to other thermodynamic properties. Doing so is particularly valuable since c_p and c_v are measurable. Unlike internal energy, enthalpy, etc., values can be experimentally determined using calorimetry.

To develop the relationships between the properties, begin with a differential form energy balance on a closed system

We will consider a system where the only thing that occurs is sensible heating, so kinetic and potential energy are negligible. Each of the two paths of interest will be considered separately.

Constant Volume

Constant Pressure

Notice that we have eliminated all inexact differentials, indicating that c_p and c_v are state properties. Therefore, these equations are valid for any substance and any process no matter what path is followed during the process.

These are the relationships that matter most. Try not to focus on the V and P in the symbols and names, since these apply only to the special cases of calculating Q. Instead try to make the links:

c_p calculates H
c_b calculates U

in your head, as these will be more valuable.

Obtaining and Using Heat Capacities

Because specific heats can be determined experimentally, many researchers have measured values for various substances. These are widely tabulated and so can be looked up in reference works.

Specific heats are functions of temperature and pressure. When you use a value, you should make sure it is valid for the range you need.

Of particular interest in this course are the limiting values of the specific heats for gases as the pressure approaches zero. We will call these zero-pressure specific heats the "ideal gas" specific heats and give them a particular notation: c_p0 and c_v0.

Tabulated values often directly reflect the temperature dependence of the specific heat by providing coefficients for a polynomial expansion:

These expansions should be used whenever the temperature range is large.

Relationships Between Specific Heats

We have looked at the relationship between specific heat and internal energy and enthalpy. The values also relate to each other, with the nature of the relationship depending on the particulars of the case. You should observe that c_p is always greater than or equal to c_v. This is because a constant pressure path will often allow the volume of the system to change, so boundary work is done, increasing the energy requirement of a constant pressure path relative to a constant volume path.

References:

Cengel, Y.A. and M.A. Boles, Thermodynamics: An Engineering Approach (3rd ed.), WCB McGraw-Hill, 1998, pp. 140-42.
Elliott, J.R. and C.T. Lira, Introductory Chemical Engineering Thermodynamics, Prentice Hall PTR, 1999, pp. 51-53, 55-56.
Moran, M.J. and H.N. Shapiro, Fundamentals of Engineering Thermodynamics (5th ed.), John Wiley, 2004, p. 98
Sonntag, R.E., C. Borgnakke, and G.J. Van Wylen, Fundamentals of Thermodynamics (6th ed.), John Wiley, 2003, pp. 133-35.
Sonntag, R.E., C. Borgnakke, and G.J. Van Wylen, Fundamentals of Thermodynamics (5th ed.), John Wiley, 1998, pp. 111-13.

R.M. Price
Original: 5/11/2000
Modified: 6/5/2000, 5/16/2003, 5/20/2004; 6/29/2005