Free Expansion

    • 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).

      3. Discuss your results.

    • assignment Adiabatic Compression

      assignment Homework

      Adiabatic Compression
      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?

    • grading Free expansion

      grading Quiz

      60 min.

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

      adiabatic expansion entropy temperature ideal gas

      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 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

      work heat first law energy

      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.
    • 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.
    • 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

      latent heat heat capacity internal energy entropy

      This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.
    • assignment Nucleus in a Magnetic Field

      assignment Homework

      Nucleus in a Magnetic Field
      Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

      Nuclei of a particular isotope species contained in a crystal have spin \(I=1\), and thus, \(m = \{+1,0,-1\}\). The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, \(E=\varepsilon\), in the state \(m=+1\) and the state \(m=-1\), compared with an energy \(E=0\) in the state \(m=0\), i.e. each nucleus can be in one of 3 states, two of which have energy \(E=\varepsilon\) and one has energy \(E=0\).

      1. Find the Helmholtz free energy \(F = U-TS\) for a crystal containing \(N\) nuclei which do not interact with each other.

      2. Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

      3. Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

    • group Heat and Temperature of Water Vapor (Remote)

      group Small Group Activity

      5 min.

      Heat and Temperature of Water Vapor (Remote)

      Thermo Heat Capacity Partial Derivatives

      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 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.
    • assignment Entropy and Temperature

      assignment Homework

      Entropy and Temperature
      Energy Temperature Ideal gas Entropy Thermal and Statistical Physics Spring 2021

      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.

  • 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?