Show that for a reversible heat pump the energy required per unit of
heat delivered inside the building is given by the Carnot
efficiency: \begin{align}
\frac{W}{Q_H} &= \eta_C = \frac{T_H-T_C}{T_H}
\end{align} What happens if the heat pump is not reversible?
Assume that the electricity consumed by a reversible heat pump must
itself be generated by a Carnot engine operating between the even
hotter temperature \(T_{HH}\) and the cold (outdoors) temperature
\(T_C\). What is the ratio \(\frac{Q_{HH}}{Q_H}\) of the heat
consumed at \(T_{HH}\) (i.e. fuel burned) to the heat delivered at
\(T_H\) (in the house we want to heat)? Give numerical values for
\(T_{HH}=600\text{K}\); \(T_{H}=300\text{K}\);
\(T_{C}=270\text{K}\).
Draw an energy-entropy flow diagram for the combination heat
engine-heat pump, similar to Figures 8.1, 8.2 and 8.4 in the text
(or the equivalent but sloppier) figures in the course notes.
However, in this case we will involve no external work at all, only
energy and entropy flows at three temperatures, since the work done
is all generated from heat.
Transitioning from the PDM back to thermodynamic systems
Heating
In the partial derivative machine, the change in internal energy corresponds to the work done on the left string and the right string:
\begin{align}
dU &= F_L dx_L + F_R dx_R
\end{align}
The ”thing we changed” was \(dx_L\) or \(dx_R\). From that we could determine the change in internal energy.
When we transfer energy to something by heating, it's hard to measure the “thing we changed,” which was entropy. It is, however, possible in some cases to measure the amount of energy transfered by heating, and from that we can work backwards to find out how much the entropy changed.
An infinitesimal amount of energy transfered by heating is called \({\mathit{\unicode{273}}} Q\). The symbol \({\mathit{\unicode{273}}} \) indicates an inexact differential, which you can think of as a “small chunk” that is not the change of something. \({\mathit{\unicode{273}}} Q\) is nota small change in the amount of energy transfered by heating, but rather is a small amount of energy transfered by heating.
When playing with the partial derivative machine, we can say the work done on the left string, \(F_Ldx_L\), is analogous to heat entering a thermodynamic system.
Latent heat
A phase transition is when a material changes state of matter, as in melting or
boiling. At most phase transitions (technically, abrupt phase transitions
as you will learnin the Capstone), the temperature remains constant while the
material is changing from one state to the other. So you know that as long as
you have ice and water coexisting in equilibrium at one atmosphere of pressure,
the temperature must be \(0^\circ\)C. Similarly, as long as water is boiling at
one atmosphere of pressure, the temperature must be \(100^\circ\)C. In both of
these cases, you can transfer energy to the system (as we will) by heating
without changing the temperature! This relates to why I keep awkwardly
saying
“transfer energy to a system by heating” rather than just “heating a system”
which means the same thing. We have deeply ingrained the idea that “heating”
is synonymous with “raising the temperature,” which does not align with the
physics meaning.
So now let me define the latent heat. The latent heat is the amount
of energy that must be transfered to a material by heating in order to change
it from one phase to another. The latent heat of fusion is the amount
of energy required to melt a solid, and the latent heat of vaporization
is the amount of energy required to turn a liquid into a gas. We will be
measuring both of these for water.
A question you may ask is whether the latent heat is extensive or intensive.
Technically the latent heat is extensive, since if you have more material
then more energy is required to melt/boil it. However, when you hear latent heat
quoted, it is almost always the specific latent heat,
which is the energy
transfer by heating required per unit of mass. It can be confusing that people
use the same words to refer to both quantities. Fortunately, dimensional checking
can always give you a way to verify which is being referred to. If \(L\) is an
energy per mass, then it must be the specific latent heat, while if it is an
energy, then it must be the latent heat.
Heat capacity and specific heat
The heat capacity is the amount of energy transfer required per
temperature to raise the temperature of a system. If we hold the pressure fixed
(as in our experiment) we can write this as:
\begin{align}
{\mathit{\unicode{273}}} Q &= C_p dT
\end{align}
where \(C_p\) is the heat capacity at fixed pressure.
You might think to rewrite this expression as a derivative, but we can't
do that since the energy transfered by heating is not a state function.
Note that the heat capacity, like the latent heat, is an extensive quantity.
The specific heat is the the heat capacity per unit mass, which is an
intensive quantity that we can consider a property of a material independently
of the quantity of that material.
I'll just mention as an aside that the term “heat capacity” is another one of
those unfortunate phrases that reflect the inaccurate idea that heat is a
property of a system.
Entropy
Finally, we can get to entropy. Entropy is the “thing that changes” when you
transfer energy by heating. I'll just give this away:
\begin{align}
{\mathit{\unicode{273}}} Q &= TdS
\end{align}
where this equation is only true if you make the change quasistatically
(see another lecture). This allows us to find the change in entropy if we know
how much energy was transfered by heating, and the temperature in the process.
\begin{align}
\Delta S &= \int \frac1T {\mathit{\unicode{273}}} Q
\end{align}
where again, we need to know the temperature as we add heat.
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.
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?
VaporizationHeat Found in: Thermal and Statistical Physics course(s)
Let's apply the relationship of heat, entropy, and temperature to a contemporary challenge!
We'd like to maximize the efficiency of any process that is based on heat flow as an input.
Just a few examples of heat engines.
Energy flow diagram
Energy flow diagram
The efficiency of the machine is
\begin{align}
\text{efficiency} &= \frac{W}{Q_{\text{in}}}
\\
\textit{e.g.} &=\frac{500\text{ J}}{1000\text{ J}} = 50\%
\end{align}
For a car engine, \(T_H\approx 600\text{ K}\) and \(T_C\approx 300\text{ K}\).
Remember that \(\Delta S=\frac{Q}{T}\), and \(\Delta S_{\text{tot}} \ge 0\).
Consider a column of atoms each of mass \(M\) at temperature \(T\) in
a uniform gravitational field \(g\). Find the thermal average
potential energy per atom. The thermal average kinetic energy is
independent of height. Find the total heat capacity per atom. The
total heat capacity is the sum of contributions from the kinetic
energy and from the potential energy. Take the zero of the
gravitational energy at the bottom \(h=0\) of the column. Integrate
from \(h=0\) to \(h=\infty\). You may assume the gas is ideal.
A black (nonreflective) sheet of metal at high
temperature \(T_h\) is parallel to a cold black sheet of metal at temperature
\(T_c\). Each sheet has an area \(A\) which is much greater than the distance between them. The sheets are in vacuum, so energy can only be transferred by radiation.
Solve for the net power transferred between the two sheets.
A third black metal sheet is
inserted between the other two and is allowed to come to a steady
state temperature \(T_m\). Find the temperature of the middle sheet,
and solve for the new net power transferred between the hot and cold sheets.
This is the principle of the heat shield, and is part of how the James Web telescope shield works.
Optional: Find the power through an \(N\)-layer sandwich.
Consider a system of
fixed volume in thermal contact with a resevoir. Show that the mean
square fluctuations in the energy of the system is \begin{equation}
\left<\left(\varepsilon-\langle\varepsilon\rangle\right)^2\right>
= k_BT^2\left(\frac{\partial U}{\partial T}\right)_{V}
\end{equation} Here \(U\) is the conventional symbol for
\(\langle\varepsilon\rangle\). Hint: Use the partition function
\(Z\) to relate \(\left(\frac{\partial U}{\partial T}\right)_V\) to
the mean square fluctuation. Also, multiply out the term
\((\cdots)^2\).
This lab gives students a chance to take data on the first day of class (or later, but I prefer to do it the first day of class). It provides an immediate context for thermodynamics, and also gives them a chance to experimentally measure a change in entropy. Students are required to measure the energy required to melt ice and raise the temperature of water, and measure the change in entropy by integrating the heat capacity.
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
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.
