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?
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.
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.
Energy and Entropy Fall 2020Energy 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}
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).
Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\)
and \(y\) are measured in meters and that \(\mu\) is measured in kilograms.
Four points are indicated on the plot.
Determine \(\frac{\partial\mu}{\partial x}\) and \(\frac{\partial\mu}{\partial
y}\) at each of the four points.
On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the
gradient of \(\mu(x,y)\) using your answers to part a above. How did you choose
a scale for your vectors? Describe how the direction of the gradient vector is
related to the contours on the plot and what property of the contour map is
related to the magnitude of the gradient vector.
Evaluate the gradient of
\(h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3\)
at the
point \((x,y)=(3,-2)\).
A contour map for a different function is shown above. On a printout of this
contour map, sketch a field vector map of the gradient of this function (sketch
vectors for at least 10 different points). The direction and magnitude of your
vectors should be qualitatively accurate, but do not calculate the gradient for
this function.
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.
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)?
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.
Energy and Entropy Fall 2020Energy 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.
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.
