At a power plant that produces 1 GW
(\(10^{9} \text{watts}\)) of electricity, the steam turbines take in steam at a
temperature of \(500^{o}C\), and the waste energy is expelled into the
environment at \(20^{o}C\).
What is the maximum possible efficiency of this plant?
Suppose you arrange the power plant to expel its waste energy
into a chilly mountain river at \(15^oC\). Roughly how much money can
you make in a year by installing your improved hardware, if you sell
the additional electricity for 10 cents per kilowatt-hour?
At what rate will the plant expel waste energy into this river?
Assume the river's flow rate is 100 m\(^{3}/\)s. By how much will
the temperature of the river increase?
To avoid this “thermal pollution” of the river the plant could
instead be cooled by evaporation of river water. This is more
expensive, but it is environmentally preferable. At what rate must
the water evaporate? What fraction of the river must be evaporated?
VaporizationHeatThermal and Statistical Physics 2020
The pressure
of water vapor over ice is 518 Pa at \(-2^\circ\text{C}\). The vapor
pressure of water at its triple point is 611 Pa, at
0.01\(^\circ\text{C}\) (see
Estimate in \(\text{J
mol}^{-1}\) the heat of vaporization of ice just under freezing. How
does this compare with the heat of vaporization of water?
Clausius-ClapeyronThermal and Statistical Physics 2020
Calculate based on the Clausius-Clapeyron equation the value of
\(\frac{dT}{dp}\) near \(p=1\text{atm}\) for the liquid-vapor
equilibrium of water. The heat of vaporization at
\(100^\circ\text{C}\) is \(2260\text{ J g}^{-1}\). Express the result
in kelvin/atm.
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.
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.
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?
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?
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?
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.
In this remote-friendly activity, students use a microwave oven (and optionally a thermometer) to measure the latent heat of melting for water (and optionally the heat capacity). From these they compute changes in entropy. See also Ice Calorimetry Lab.
Energy and Entropy 2021 (2 years)
Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
Find the partial derivatives
\[\left(\frac{\partial {S}}{\partial {T}}\right)_{p}\]
\[\left(\frac{\partial {S}}{\partial {T}}\right)_{V}\]
where \(p\) is the pressure, \(V\) is the volume, \(S\) is the entropy, and \(T\) is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
Why does it take only two variables to define the state?
Why are the derivatives above different?
What do the words isobaric, isothermal, and isochoric mean?
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, from which they compute changes in entropy.
You are given the following Gibbs free energy:
\begin{equation*}
G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right)
\end{equation*}
where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).
Compute the entropy.
Work out the heat capacity at constant pressure \(C_p\).
Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).
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}
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 \(U=\frac32 Nk_BT\).
How many molecules of gas does the large bottle contain? What is the final temperature of the gas?
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).
Discuss your results.
Energy and Entropy 2021 (2 years)
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.
What is the maximum possible efficiency of an engine operating
between these two temperatures?
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.
