Lecture T6: Cv, Cp

I started the lecture with a discussion of cp and cv. The Dmud was large on this concept (45 people unclear after T5, 17 after T6) which I am happy to see. I concluded this discussion with two PRS questions about the relative magnitudes of cv and cp for gases and solids. These questions were intended to underscore several important points: the definition of cp and cv, the balances between heat and work and energy change in two important processes--constant volume and constant pressure, and the distinction between the definition of cp and cv and their broader applicability in cases of ideal gases. We then did a third PRS that went along with the experiment to heat a cup of water. The purpose of this PRS was to provide a first example of the application of specific heat to solve a problem and to give you some practice estimating the performance of a real process.

Responses to 'Muddiest Part of the Lecture Cards'

(49 respondents out of 67 students in class)

1) I am confused on cv and cp (5 students). First read this again (along with the notes). Now let me add a few of the more subtle points. The quantities u, h, and T are all properties of the system and thus are functions of the thermodynamic state of the system (not the manner in which you got there). So the expressions du=cvdT and dh=cpdT are valid at any point in any ideal gas process (it doesn't have to be constant volume or constant pressure) as long as it is possible to define the state of the system. (That is, you must at least be able to assume that it is in quasi-thermodynamic equilibrium at the point you are considering.)

2) I am still a bit confused as to why cv<cp (even after your explanation) (3 students). First, it is only true for a gas. For solids and most liquids cp is approximately equal to cv. The difference is that gases are compressible--that is, they change their specific volume relatively easily. So when we add heat to a gas it will expand unless we constrain it. The expansion takes some of the energy away from internal energy to do work on the surroundings. When we constrain the volume of the gas and do not allow it to expand, it therefore takes less heat addition to increase its temperature by 1K, so cv<cp.

3) If cp is dereived in terms of enthalpy, why can it be used to preduct delta-u? (1 student) First we didn't derive it, we defined. Second, cp is used with changes in temperature to find delta-h for an ideal gas, not delta-u.

4) Do you need the small subscipts on the partial derivatives? Don't they imply everything else is held constant? (1 student) Yes they do. I just add the subscripts to make it a little clearer.

5) It seems strange that du=cvdT holds for any process of an ideal gas. (2 students) This is based on observations of the world around us. Strange or not, it is very convenient because it makes it much easier to solve problems.

6) Even though we are assuming constant specific heats for the purposes of the class, what does the function used to get an accurate figure look like? (1 student) If you look in the back of Sonntag, Bornakke and Van Wylen (on reserve) you will see that these properties are tabulated for many materials and gases. Cv and Cp are usually expressed as 3rd or 4th order polynomials in temperature.

7) What happens with cp and cv for a non-constant pressure or volume process? (1 student) For an ideal gas they can be used to relate the change in temperature to the changes in enthalpy and internal energy for any process. For non-ideal gases, additional properties must be known.

8) I missed the difference between the symbols, del, delta, and d. (2 students) The del denotes a partial derivative (change with respect to one variable only even if the quantity is a function of more than one variable. The "d" denotes a total derivative -- the total change with respect to all of the variables. The delta I use to denote a small change in a path dependent variable (like q and w); I don't use a "d" because they are not exact differentials.

9) What did the statement "U not equal to U(T,V) for many gases" mean in your derivations for cv? (1 student) For ideal gases (many of the gases around us frequently behave this way), U=U(T) only.

10) Could you go over how you made the estimate for the experiment ? (4 students) Please read the discussion I put on the answer button.

11) I still don't understand the units in PV=mRT. (1 student) I think you will become more confortable after using it to solve some problems. If not, please see me and we can talk abooout it.

12) How is enthalpy useful? (2 students) You will see over the next several lectures that it is convenient for many problems.

13) If you instantaneously remove weights from a piston, would the expansion of the gas be an isothermal process? (1 student) It depends on how much work is done and how much heat is transferred. If they balance so that the internal energy is constant, then it will be isothermal.

14) Can you give us more tips on estimation and such? (1 student) I will. Estimating is a very important skill for engineers.

10) No mud (23 students). Good.