.. _lecture4:
Lecture 4: First Law of Thermodynamics
++++++++++++++++++++++++++++++++++++++
.. note::
*Classical thermodynamics ... is the only physical theory of universal content which I am convinced ... will never be overthrown.* — Albert Einstein
.. warning::
This lecture corresponds to Chapters 11 and 12 of the textbook.
Summary
-------
.. attention::
In this lecture, we introduce the second law of thermodynamics and find that changes in internal energy can take place by a transfer of heat or work in or out of the system. Formally:
.. math::
dU = \dbar Q + \dbar W
The notation above is loaded, and it is important to understand the
"d-bar" we used there. A straight "d" refers to an exact
differential, that is: its integration only depends on the starting
and end points (this is called a **function of state**, since it
only depends on the state rather than **how** we got
there). Conversely, the "d-bar" refers to an inexact differential,
that is: its integral between two states **depends on the path
taken**. It follows that :math:`Q` and :math:`W` are not functions of state (that's
why we never actually use :math:`Q` or :math:`W`, and we use their changes instead). This is
a crucial point that governs classical thermodynamics. Students
should really spend some time making sure they understand this
crucial point! More information can be found in :ref:`exactdiff`.
In this lecture, we also studied two thermodynamic processes in
greater details: the adiabatic expansion and the isothermal
expansion of an ideal gas (see below for a definition of those
terms). We were able to come up with two extremely important
results that describe how to process translates in the
:math:`p vs. V` phase diagram during the process.
.. image:: _images/expansionsNBG.png
:width: 350
:alt: Expansions
:class: with-shadow float-left
For an **isothermic process**: :math:`pV=\textrm{constant}` (the constant grows with temperature).
For an **adiabatic process**: :math:`pV^{\gamma}=\textrm{constant}` where :math:`\gamma=c_p/c_V` is a constant larger than one (it is 5/3 for an ideal gas).
Note also that during an adiabatic expansion, the temperature
decreases (the reason is simple: during an expansion, work is done
**by** the system, so it is negative; as a result, the total energy
:math:`U` decreases and since, for an ideal gas, the internal
energy is proportional to temperature, the temperature must
decrease). More information on the sign convention can be found below.
During an isothermal expansion, for the same reason as the
one given in the previous paragraph, the total energy is
constant. This means that the work done by the system is negative
(i.e., work done by the engine) and is exactly equal (in absolute value)
to the heat transferred from the reservoir (the heat transfer is
positive since it is transferred **to** the system, using our
selfish sign convention).
.. note::
**Sign convention**
In this course, we will need to be careful about the sign of the
heat transfers and work done. We will employ the usual :index:`selfish
sign convention`: :math:`\Delta Q` will be positive when heat is added to
a system and :math:`\Delta W` will be positive when it is work done
on a system.
It will happen that a different convention will be used (especially
in :ref:`lecture5`) where the positive sign will correspond to the
arrows drawn on the figure. This should not cause any particular
issue.
Learning Material
-----------------
Copy of Slides
~~~~~~~~~~~~~~
The slides for Lecture 4 are available in pdf format here: :download:`pdf <_pdfs/slides/lecture4.pdf>`
Screencast
~~~~~~~~~~
.. raw:: html
.. raw:: latex
This lecture is available as two YouTube recordings available at these links: \href{https://www.youtube.com/embed/1gV50WNr5yM}{chapter 11}. and \href{https://www.youtube.com/embed/WsKcohkC_jA}{chapter 12}.
Key Definitions
---------------
.. note::
:index:`Thermal equilibrium`:
A system is in thermal equilibrium
when its macroscopic observables (such as its pressure or its
temperature) have ceased to change with time.
:index:`Function of state`:
any physical quantity that has a well
defined value for each equilibrium state of the
system. Mathematically, a function of state corresponds to an exact
differential.
:index:`Exact differential`:
function whose integration between two
given points does not depend on the path taken between the
points. This definition is to be contrasted with the notion of
:index:`inexact differential`. Important examples of inexact
differentials are: heat and work.
:index:`Internal energy`:
is the sum of the energy of all the
internal degrees of freedom that the system possesses. This is a
function of state.
:index:`Thermally isolated` system:
a physical system that cannot exchange heat with its surroundings.
:index:`Reversible work`:
work performed on a gas at pressure :math:`p` is :math:`\dbar W=-p \mathrm{~d} V`.
:index:`Adiabatic index`:
(:math:`\gamma`) is the ratio between the
heat capacity at constant pressure and the heat capacity at
constant volume. It is always larger than one, and 5/3 for an ideal
gas.
:index:`Reversible process`: A process that takes place
quasi-statically and for which the systems remain in equilibrium
throughout the process. In reality, no actual process can be
strictly reversible.
:index:`Isothermal expansion` of a gas: an expansion taking place
in contact of a heat reservoir (i.e., at constant temperature).
:index:`Adiabatic expansion` of a gas: an expansion taking place in
a thermally isolated container (i.e., without heat exchange with
the surroundings).
A full list of terms, including the ones provided here, can be found in the :ref:`genindex`.
Test your knowledge
-------------------
1. What is a function of state?
A. A function of state is a function that describes the thermodynamic evolution of two systems in contact.
B. A function of state is a function that describes the state of a system, regardless how the system achieved it.
C. A function of state is a function that describes the state of a system, provided one knows how the system reached that state. distribution as a function of temperature.
D. None of the other options is correct.
2. The first law of thermodynamics states that:
A. Heat and work are two forms of energy that can be transformed from one into the other. The sum of the two types is constant.
B. Total energy is a conserved quantity, though work and heat contributions are not individually conserved.
C. Total energy is a function of state, work and heat are not.
D. All the provided definitions are correct.
3. A thermally isolated system...
A. Has constant temperature.
B. Has constant total energy.
C. Cannot exchange heat with its surrounding.
D. More than one option are correct.
E. None of the other options is correct.
4. The internal energy of an ideal gas...
A. Is uniquely defined if one knows its temperature.
B. Is uniquely defined if one knows its temperature and the volume it occupies.
C. Is dominated by internal interaction energy.
D. None of the other options is correct.
5. Compare the expansion of an ideal gas performed adiabatically and that performed isothermally. Consider the :math:`p-V` evolution.
A. :math:`p(V)` decays faster for the adiabatic process compared to the isothermal one.
B. :math:`p(V)` decays slower for the adiabatic process compared to the isothermal one.
C. The decay of :math:`p(V)` is always irreversible, even for a quasi-static expansion.
D. None of the other options is correct.
.. hint::
Find the answer keys on this page: :ref:`answerkeys`. Don't cheat! Try solving the problems on your own first!
Homework Assignment
-------------------
Solve the following problems from the textbook: 2.5, 3.4, 4.2, 4.3, 4.4