Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
That thermal system have at least two independent variables. You have to specify changes in both to determine how other (dependent) quantities will change.
Media
Goals
For functions of two independent variables, you need to specify changes in both independent variables to determine how the function changes.
Can lead into a lecture about partial derivatives and specifying a path.
Time Estimate: 15 minutes
Tools/Equipment
A Purple \(U(S,V)\) Plastic Surface for each group
Dry-erase markers
Intro
Students need to be oriented to the surfaces and what they represent.
Whole Class Discussion:
The main take-away is that for functions of two variables, you need to specify a change in both the independent variables to know the change in the function.
You can ask student groups to share an example of a path where the volume increases and the internal energy: increases, decreases, and is constant.
This activity leads into ideas about (1) how the number of independent variable you have in a thermal system corresponds to the number of ways of getting energy into or out of a system, (2) you need to specify the path for a partial derivative, and (3) for thermal system, you actually CANNOT “hold all the other variables constant”.
Orient: The surface represents measurements of internal energy on a kilogram of water vapor in a piston (a graduated cylinder with a moveable top). The purple surface graph is \(U(S,V)\).
Interpret: Starting at the red star, as you increase the volume of the system, what happens to the internal energy of the water vapor?
Answer: The internal energy could increase, decrease, or stay the same. It depends on how the volume is increasing (and what is happening to the entropy).
Student Ideas: Most students will interpret “increasing volume” as “hold the entropy fixed”. This is, of course, a possibility, but it is not the only possible way to increase the volume. From their math classes, many students will have the idea that for a partial derivative, everthing else must be held constant.
Discussion: Could you use the other surface? If so, how?
Discussion: Does your answer depend on the initial state of the system?}
Clarification: Students may need guidance to recognize the meaning of the contour lines, and the fact that they can use either surface to answer this question (although one is much easier that the other).
Discussion: What would the surface look like if your answer did not depend on the state of the system?
Consider a Special Case: Is it possible to change the volume without changing the internal energy? (Is your answer consistent with your answer to the previous prompt?)
Answer: Yes - you move along a level curve of the internal energy. Along this level curve, both volume and entropy change.
Discussion: Some students will happily answer “internal energy increases” to the first prompt and “yes it is possible to change the volume and not change the internal energy” to the second prompt without resolving the inconsistency between the two answers.
Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.
In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
Consider one mole of an
ideal monatomic gas at 300K and 1 atm. First, let the gas expand
isothermally and reversibly to twice the initial volume; second, let
this be followed by an isentropic expansion from twice to four times
the original volume.
How much heat (in joules) is added to the gas in each of these two
processes?
What is the temperature at the end of the second process?
Suppose the first process is replaced by an irreversible expansion
into a vacuum, to a total volume twice the initial volume. What is
the increase of entropy in the irreversible expansion, in J/K?
A diesel
engine requires no spark plug. Rather, the air in the cylinder is
compressed so highly that the fuel ignites spontaneously when
sprayed into the cylinder.
In this problem, you may treat air as an ideal gas, which satisfies
the equation \(pV = Nk_BT\). You may also use the property of an
ideal gas that the internal energy depends only on the temperature
\(T\), i.e. the internal energy does not change for an isothermal
process. For
air at the relevant range of temperatures the heat capacity at
fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy
is given by \(U=\frac52Nk_BT\).
Note: in this problem you are expected to use only the equations
given and fundamental physics laws. Looking up the formula in a textbook
is not considered a solution at this level.
If the air is initially at room temperature
(taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\)
of the original volume, what final temperature is attained (before fuel
injection)?
Consider the bottle in a bottle problem in a previous problem set, summarized here.
A small bottle of
helium is placed inside a large bottle, which otherwise contains
vacuum. The inner bottle contains a slow leak, so that the helium
leaks into the outer bottle. The inner bottle contains one tenth
the volume of the outer bottle, which is insulated.
The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).
How many molecules of gas does the large bottle contain? What is the final temperature of the gas?
Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).
Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.
The internal energy is of any ideal gas can be written as
\begin{align}
U &= U(T,N)
\end{align}
meaning that the internal energy depends only on the number of
particles and the temperature, but not the volume.* The ideal gas law
\begin{align}
pV &= Nk_BT
\end{align}
defines the relationship between \(p\), \(V\) and \(T\). You may take the
number of molecules \(N\) to be constant. Consider the free adiabatic
expansion of an ideal gas to twice its volume. “Free expansion”
means that no work is done, but also that the process is also
neither quasistatic nor reversible.
What is the change in entropy of the gas? How do you know
this?
Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.
