## Student handout: 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.
What students learn
• Both heat and work contribute to changing internal energy.
• Working corresponds to volume changing (if volume is constant, no work is done). Heating means entropy is changing (if entropy is constant, no heat is transferred).
• Thermal systems follow paths in state space---different quasistatic processes follow different paths.
• The internal energy of the system can change in various ways (increase, decrease, stay the same)---the change varies by process.
• The amount of change generally depends on the initial state.
• 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.
• group Ideal Gas Model

group Small Group Activity

30 min.

##### Ideal Gas Model

Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy.
• 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.
• group Quantifying Change (Remote)

group Small Group Activity

30 min.

##### Quantifying Change (Remote)

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
• group Squishability'' of Water Vapor (Contour Map)

group Small Group Activity

30 min.

##### “Squishability” of Water Vapor (Contour Map)

Students determine the “squishibility” (an extensive compressibility) by taking $-\partial V/\partial P$ holding either temperature or entropy fixed.
• face Energy and heat and entropy

face Lecture

30 min.

##### Energy and heat and entropy
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.
• group Charged Sphere

group Small Group Activity

30 min.

##### Charged Sphere

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
• group Electric Potential of Two Charged Plates

group Small Group Activity

30 min.

##### Electric Potential of Two Charged Plates
Students examine a plastic "surface" graph of the electric potential due to two changes plates (near the center of the plates) and explore the properties of the electric potential.
• assignment_ind Partial Derivatives from a Contour Map

assignment_ind Small White Board Question

10 min.

##### Partial Derivatives from a Contour Map
AIMS Maxwell AIMS 21 Students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?

Changing Internal Energy: Consider a thermal system of water vapor where the only ways to change the internal energy are by heating or doing work:

$\mbox{change in internal energy} = \mbox{heat into or out of system}+\mbox{work done on or by system}$

For very small changes in internal energy, the above energy conservation statement turns into: \begin{eqnarray*} \mbox{small change of internal energy} &=& \mbox{small amount of heating} + \mbox{small amount of work done} \\[8pt] d\mbox{internal energy}&=&\mbox{(temperature)} d\mbox{entropy}-\mbox{(pressure)} d\mbox{volume} \\[8pt] dU&=& T\;dS - p\;dV \end{eqnarray*}

where“$d\mbox{quantity}$" indicates a very small change in a quantity.

Examine Your Intuition: For each of the three situations below, how would you expect the internal energy of the water vapor to change (increase, decrease, or stay the same)? Explain your reasoning.

1. You fill a lidded metal pot with water vapor and put it on a hot stove.

2. You fill an insulated piston (so that no heat enters or leaves) with water vapor and you push down on the lid of the piston.

3. You fill an uninsulated metal piston with a heavy top (which fixes the pressure) with water vapor and put it on a hot stove.

Interpret the Surface: The plastic surface model is a graph of the internal energy as a function of volume and entropy. The arrows in the base of the surface indicate the direction of increase for each of these quantities. The height of the surface represents the value of the internal energy.

For each of the scenarios described above, imagine that the water vapor starts with values of volume and entropy that correspond to the blue dot. For each scenario:

1. What does the surface tell you about the change in internal energy? Is this consistent with your intuition?

2. Describe in words how you determined this information from the surface.

3. Would the change in internal energy be different if you started somewhere other than the blue dot?

Author Information
Raising Physics to the Surface
Keywords
Thermo Internal Energy 1st Law of Thermodynamics
Learning Outcomes