Activity: Heat and Temperature of Water Vapor (Remote)

• group Small Group Activity schedule 5 min. build Heat Capacity graph, Student handout for each student, A personal or shared writing space for each student to write/draw/sketch. description Student handout (PDF)
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  Thermo Heat Capacity Partial Derivatives

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  • group Number of Paths

    group Small Group Activity

    30 min.

    Number of Paths

    E&M Conservative Fields Surfaces

    Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.
  • group Calculating Coefficients for a Power Series

    group Small Group Activity

    30 min.

    Calculating Coefficients for a Power Series

    AIMS Maxwell AIMS 21 Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

    Taylor series power series approximation

    Power Series Sequence (E&M)

    This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:

    \[c_n={1\over n!}\, f^{(n)}(z_0)\]

    After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a \(\sin\theta\) (around both the origin and \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.

  • group Static Fields Equation Sheet

    group Small Group Activity

    5 min.

    Static Fields Equation Sheet

    Static Fields Winter 2021 Static Fields Winter 2021
  • assignment Isothermal/Adiabatic Compressibility

    assignment Homework

    Isothermal/Adiabatic Compressibility

    Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

    The isothermal compressibility is defined as \begin{equation} K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T} \end{equation} \(K_T\) is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as \begin{equation} K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S} \end{equation} and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that \begin{equation} \frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}} \end{equation} Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

  • assignment Power from the Ocean

    assignment Homework

    Power from the Ocean

    heat engine efficiency Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

    It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is \(22^\circ\)C at the ocean surface and \(4^{o}\)C at the ocean floor.

    1. What is the maximum possible efficiency of an engine operating between these two temperatures?

    2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water \(c_p = 4.2\) Jg\(^{-1}\)K\(^{-1}\) and the density of water is 1 g cm\(^{-3}\), and both are roughly constant over this temperature range.

  • group Changes in Internal Energy (Remote)

    group Small Group Activity

    30 min.

    Changes in Internal Energy (Remote)

    Thermo Internal Energy 1st Law of Thermodynamics

    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 Covariation in Thermal Systems

    group Small Group Activity

    30 min.

    Covariation in Thermal Systems

    Thermo Multivariable Functions

    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 Ice calorimetry lab questions

    assignment Homework

    Ice calorimetry lab questions

    This question is about the lab we did in class: Ice Calorimetry Lab.

    1. Plot your data I Plot the temperature versus total energy added to the system (which you can call \(Q\)). To do this, you will need to integrate the power. Discuss this curve and any interesting features you notice on it.
    2. Plot your data II Plot the heat capacity versus temperature. This will be a bit trickier. You can find the heat capacity from the previous plot by looking at the slope. \begin{align} C_p &= \left(\frac{\partial Q}{\partial T}\right)_p \end{align} This is what is called the heat capacity, which is the amount of energy needed to change the temperature by a given amount. The \(p\) subscript means that your measurement was made at constant pressure. This heat capacity is actually the total heat capacity of everything you put in the calorimeter, which includes the resistor and thermometer.
    3. Specific heat From your plot of \(C_p(T)\), work out the heat capacity per unit mass of water. You may assume the effect of the resistor and thermometer are negligible. How does your answer compare with the prediction of the Dulong-Petit law?
    4. Latent heat of fusion What did the temperature do while the ice was melting? How much energy was required to melt the ice in your calorimeter? How much energy was required per unit mass? per molecule?
    5. Entropy of fusion The change in entropy is easy to measure for a reversible isothermal process (such as the slow melting of ice), it is just \begin{align} \Delta S &= \frac{Q}{T} \end{align} where \(Q\) is the energy thermally added to the system and \(T\) is the temperature in Kelvin. What is was change in the entropy of the ice you melted? What was the change in entropy per molecule? What was the change in entropy per molecule divided by Boltzmann's constant?
    6. Entropy for a temperature change Choose two temperatures that your water reached (after the ice melted), and find the change in the entropy of your water. This change is given by \begin{align} \Delta S &= \int \frac{{\mathit{\unicode{273}}} Q}{T} \\ &= \int_{t_i}^{t_f} \frac{P(t)}{T(t)}dt \end{align} where \(P(t)\) is the heater power as a function of time and \(T(t)\) is the temperature, also as a function of time.
  • biotech Microwave oven Ice Calorimetry Lab

    biotech Experiment

    60 min.

    Microwave oven Ice Calorimetry Lab

    Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

    heat entropy water ice thermodynamics

    The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply.
  • group Ice Calorimetry Lab

    group Small Group Activity

    60 min.

    Ice Calorimetry Lab

    heat entropy water ice

    The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply.