## Activity: Ideal Gas Model

Students consider whether the thermo surfaces reflect the properties of an ideal gas.
• group Small Group Activity schedule 30 min. build Ideal Gas Contour Maps, handout for each student, purple and blue thermo surfaces for instructor, personal or shared writing space for each student description Student handout (PDF)
• Search for related topics
Ideal Gas surfaces thermo
What students learn
• Equation and definition of an Ideal Gas
• Students explore what it would mean for something to be an ideal gas or a good approximation of an ideal gas
• Students might discuss the difference between temperature and internal energy
• Can discuss degrees of freedom
• Media

The internal energy of an ideal gas is described by the equation:

$U(p,T)=\left(\frac{\# dof}{2}\right) NkT$

The surfaces represent the internal energy of water vapor:

• The blue surface shows the internal energy as a function of temperature and pressure, $U(T,p)$, with volume and entropy contours etched into the surface.
• The purple surface shows the internal energy as a function of volume and entropy, $U(S,V)$, with pressure and temperatures contours etched into the surface.

By examining the plastic surfaces, how can you tell if an ideal gas model is a good model for water vapor?

Answer: The biggest indicator on the p-T surface is if the surface is a plane. On the contour map, this corresponds to the internal energy contours being nearly parallel to the volume axis (constant at constant temperature) AND having the same, constant spacing between them. Also, volume contours are nearly linear (ideal gas law). For an ideal gas on the S-V graph, isotherms should be parallel to and never cross the U contours, i.e. on the purple surface the isotherms are level curves.

Discussion: Sketching a 2D graph of U vs T for an Ideal Gas When working with a contour map, students might benefit from sketching a graph of internal energy vs.temperature for an ideal gas, then sketching a plot of internal energy vs.temperature for water vapor, using the values from the contour map.

Discussion: Linearity The instructor might model the blue surface, describing the features of the surface, including its near-flatness. (Although it is not a perfectly flat, and this can be demonstrating by placing the surface upside down on a flat table, against a small whiteboard, etc.)

Discussion: Applications of the Model One can discuss degrees of freedom and gasses/situations that are typically modeled as ideal.

If you give the students the scaling and the number of particles, they could be able to figure out the approximate degrees of freedom for water vapor (canonically, 9).
• 1kg water vapor is 55 mols of water
•  Internal Energy 2cm $\rightarrow$ 170. kJ Temperature 2cm $\rightarrow$ 70 K

SUMMARY PAGE
Goals:
• Equation and definition of an Ideal Gas
• Students explore what it would mean for something to be an ideal gas or a good approximation of an ideal gas
• Students might discuss the difference between temperature and internal energy
• Can discuss degrees of freedom

Time Estimate: 15-30 minutes

Tools and Equipment:

• Purple and Blue Thermo surfaces for each group
• Remote Option: Ideal Gas Contour Maps
• Student handout for each student
• A personal or shared writing space for each student to write/draw/sketch

Intro:

• Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy

Whole Class Discussion:

• A whole class discussion should elicit student ideas about how to determine if a system fits the ideal gas model perfectly, and when the ideal gas model is a “good" fit.
• Compare and contrast how the ideal gas relationships appear on the blue vs purple surface. In particular, U $\propto$ T - blue surface is a plane; on the purple (and blue) surface the temperature and internal energy contours are parallel.

• group Heat and Temperature of Water Vapor

group Small Group Activity

30 min.

##### Heat and Temperature of Water Vapor

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.
• group Covariation in Thermal Systems

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

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.
• group Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

##### Changes in Internal Energy (Remote)

Warm-Up

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.
• assignment Ideal gas calculations

assignment Homework

##### Ideal gas calculations
Ideal gas Entropy Sackur-Tetrode Thermal and Statistical Physics 2020

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.

1. How much heat (in joules) is added to the gas in each of these two processes?

2. What is the temperature at the end of the second process?

3. 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?

• assignment Free Expansion

assignment Homework

##### Free Expansion
Energy and Entropy 2021 (2 years)

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.
1. What is the change in entropy of the gas? How do you know this?

2. What is the change in temperature of the gas?

assignment Homework

ideal gas internal energy engine Energy and Entropy 2020

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.

1. 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)?

2. By what factor does the pressure increase?

• assignment Centrifuge

assignment Homework

##### Centrifuge
Centrifugal potential Thermal and Statistical Physics 2020 A circular cylinder of radius $R$ rotates about the long axis with angular velocity $\omega$. The cylinder contains an ideal gas of atoms of mass $M$ at temperature $T$. Find an expression for the dependence of the concentration $n(r)$ on the radial distance $r$ from the axis, in terms of $n(0)$ on the axis. Take $\mu$ as for an ideal gas.
• assignment Active transport

assignment Homework

##### Active transport
Active transport Concentration Chemical potential Thermal and Statistical Physics 2020

The concentration of potassium $\text{K}^+$ ions in the internal sap of a plant cell (for example, a fresh water alga) may exceed by a factor of $10^4$ the concentration of $\text{K}^+$ ions in the pond water in which the cell is growing. The chemical potential of the $\text{K}^+$ ions is higher in the sap because their concentration $n$ is higher there. Estimate the difference in chemical potential at $300\text{K}$ and show that it is equivalent to a voltage of $0.24\text{V}$ across the cell wall. Take $\mu$ as for an ideal gas. Because the values of the chemical potential are different, the ions in the cell and in the pond are not in diffusive equilibrium. The plant cell membrane is highly impermeable to the passive leakage of ions through it. Important questions in cell physics include these: How is the high concentration of ions built up within the cell? How is metabolic energy applied to energize the active ion transport?

You might wonder why it is even remotely plausible to consider the ions in solution as an ideal gas. The key idea here is that the ideal gas entropy incorporates the entropy due to position dependence, and thus due to concentration. Since concentration is what differs between the cell and the pond, the ideal gas entropy describes this pretty effectively. In contrast to the concentration dependence, the temperature-dependence of the ideal gas chemical potential will not be so great.

• face Work, Heat, and cycles

face Lecture

120 min.

##### Work, Heat, and cycles
Thermal and Statistical Physics 2020

These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.
• assignment Photon carnot engine

assignment Homework

##### Photon carnot engine
Carnot engine Work Energy Entropy Thermal and Statistical Physics 2020

In our week on radiation, we saw that the Helmholtz free energy of a box of radiation at temperature $T$ is \begin{align} F &= -8\pi \frac{V(kT)^4}{h^3c^3}\frac{\pi^4}{45} \end{align} From this we also found the internal energy and entropy \begin{align} U &= 24\pi \frac{(kT)^4}{h^3c^3}\frac{\pi^4}{45} V \\ S &= 32\pi kV\left(\frac{kT}{hc}\right)^3 \frac{\pi^4}{45} \end{align} Given these results, let us consider a Carnot engine that uses an empty metalic piston (i.e. a photon gas).

1. Given $T_H$ and $T_C$, as well as $V_1$ and $V_2$ (the two volumes at $T_H$), determine $V_3$ and $V_4$ (the two volumes at $T_C$).

2. What is the heat $Q_H$ taken up and the work done by the gas during the first isothermal expansion? Are they equal to each other, as for the ideal gas?

3. Does the work done on the two isentropic stages cancel each other, as for the ideal gas?

4. Calculate the total work done by the gas during one cycle. Compare it with the heat taken up at $T_H$ and show that the energy conversion efficiency is the Carnot efficiency.

Author Information
Raising Physics to the Surface
Keywords
Ideal Gas surfaces thermo
Learning Outcomes