## Activity: Ideal Gas Model

Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy.
• 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)
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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
• group Heat and Temperature of Water Vapor (Remote)

group Small Group Activity

5 min.

##### Heat and Temperature of Water Vapor (Remote)

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.
• assignment Free Expansion

assignment Homework

##### Free Expansion
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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 Fall 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 Bottle in a Bottle 2

assignment Homework

##### Bottle in a Bottle 2
heat entropy ideal gas Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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 $U=\frac32 Nk_BT$.

1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

2. 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).

• assignment Entropy and Temperature

assignment Homework

##### Entropy and Temperature
Energy Temperature Ideal gas Entropy Thermal and Statistical Physics Spring 2020

Suppose $g(U) = CU^{3N/2}$, where $C$ is a constant and $N$ is the number of particles.

1. Show that $U=\frac32 N k_BT$.

2. Show that $\left(\frac{\partial^2S}{\partial U^2}\right)_N$ is negative. This form of $g(U)$ actually applies to a monatomic ideal gas.

• group Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

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

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.
• group Using $pV$ and $TS$ Plots

group Small Group Activity

30 min.

##### Using $pV$ and $TS$ Plots
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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.

60 min.

##### Free expansion
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

Students will determine the change in entropy (positive, negative, or none) for both the system and surroundings in three different cases. This is followed by an active whole-class discussion about where the entropy comes from during an irreversible process.
• group Thermodynamic States (Remote)

group Small Group Activity

30 min.

##### Thermodynamic States (Remote)

Little is needed. Some students might be bothered by thinking about entropy if it hasn't been mentioned at all in class. Try doing this activity as a follow-up to the “Changes in Internal Energy" about the first law of thermodynamics.

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: 30 minutes + 10 minutes whole class discussion

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 eleicit student ideas about how to determine if a system fits the ideal gas model perfectly, and when the ideal gas moadel is a “good" fit.

## Ideal Gas Model

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

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

The contours map presents the internal energy of water vapor: it shows volume and internal energy contours plotted on temperature and pressure axes.

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

Solution: The biggest indicator on the p-T graph is if the internal energy contours are nearly parallel to the volume axis (constant at constant temperature) AND they have 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.

Hint: 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.

WCD: 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 modeled by placing the surface upside down on a flat table, against a small whiteboard, etc.)

WCD: One can discuss degrees of freedom and gasses/situations that are typically modeled as ideal.

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