Student handout: Changes in Internal Energy

None 2023
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.

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 scenarios 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 have a lidded metal pot filled water vapor and put it on a hot stove.

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

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

Interpret the Contour Map & Plastic Graph: The plastic surface model is a graph of the internal energy of water vapor 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. The contour map corresponds to the plastic graph.


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