The
law of conservation of energy is an empirical
law of physics. It states that the total amount of
energy in an
isolated
system remains constant over time (is said to be
conserved over time). A consequence of this law is that
energy cannot be created nor destroyed. The only thing that can
happen to energy in a
closed system is
that it can change form, for instance
chemical energy can become
thermal energy.
Albert Einstein's
theory of
relativity shows that energy and mass are the same thing, and
that neither one appears without the other. Thus in closed systems,
both mass and energy are conserved separately, just as was
understood in pre-relativistic physics. The new feature of
relativistic physics is that "matter" particles (such as those
constituting atoms) could be converted to non-matter forms of
energy, such as light; or kinetic and potential energy (example:
heat). However, this conversion does
not affect
the total mass of systems, since the latter forms of non-matter
energy still retain their mass through any such conversion.
Today, conservation of “energy” refers to the conservation of the
total system energy over time. This energy includes the energy
associated with the rest mass of particles and all other forms of
energy in the system. In addition the
invariant mass of systems of particles (the
mass of the system as seen in its
center
of mass inertial frame, such as the frame in which it would
need to be weighed), is also conserved over time for any single
observer, and (unlike the total energy) is the same value for all
observers. Therefore, in an isolated system, although matter
(particles with rest mass) and "pure energy" (heat and light) can
be converted to one another, both the total amount of energy and
the total amount of mass of such systems remain constant over time,
as seen by any single observer. If energy in any form is allowed to
escape such systems (see
binding
energy) the mass of the system will decrease in correspondance
with the loss.
A consequence of the law of energy conservation is that
perpetual motion machines can only work
perpetually if they deliver no energy to their surroundings. If
such machines produce more energy than is put into them, they must
lose mass and thus eventually disappear over perpetual time, and
are therefore impossible.
History
Ancient philosophers as far back as
Thales of Miletus had inklings of the conservation of
which everything is made. However, there is no particular reason to
identify this with what we know today as "mass-energy" (for
example, Thales thought it was water). In 1638,
Galileo published his analysis of several
situations—including the celebrated "interrupted pendulum"—which
can be described (in modern language) as conservatively converting
potential energy to kinetic energy and back again. It was
Gottfried Wilhelm Leibniz during
1676–1689 who first attempted a mathematical formulation of the
kind of energy which is connected with
motion (kinetic
energy). Leibniz noticed that in many mechanical systems (of
several
masses,
m_{i} each
with
velocity v_{i} ),
- \sum_{i} m_i v_i^2
was conserved so long as the masses did not interact. He called
this quantity the
vis viva or
living force of the system. The principle represents an
accurate statement of the approximate conservation of
kinetic energy in situations where there is
no friction. Many
physicists at that time
held that the
conservation of
momentum, which holds even in systems with friction, as defined
by the
momentum:
- \,\!\sum_{i} m_i v_i
was the conserved
vis viva. It was later shown that, under
the proper conditions, both quantities are conserved simultaneously
such as in
elastic
collisions.
It was largely
engineers such as
John Smeaton,
Peter
Ewart,
Karl Hotzmann,
Gustave-Adolphe Hirn and
Marc Seguin who objected that conservation of
momentum alone was not adequate for practical calculation and who
made use of
Leibniz's principle.
The principle was also championed by some
chemists such as
William Hyde Wollaston. Academics
such as John Playfair were quick to point out that kinetic energy
is clearly not conserved. This is obvious to a modern analysis
based on the
second law of
thermodynamics but in the 18th and 19th centuries, the fate of
the lost energy was still unknown. Gradually it came to be
suspected that the
heat inevitably generated by
motion under friction, was another form of
vis viva. In
1783,
Antoine Lavoisier and
Pierre-Simon Laplace reviewed
the two competing theories of
vis viva and
caloric theory.
Count Rumford's 1798 observations of heat
generation during the
boring
of
cannons added more weight to the view that
mechanical motion could be converted into heat, and (as
importantly) that the conversion was quantitative and could be
predicted (allowing for a universal conversion constant between
kinetic energy and heat).
Vis viva now started to be known
as
energy, after the term was first used in that sense by
Thomas Young in 1807.
The recalibration of
vis viva to
- \frac {1} {2}\sum_{i} m_i v_i^2
which can be understood as finding the exact value for the kinetic
energy to
work conversion
constant, was largely the result of the work of
Gaspard-Gustave Coriolis and
Jean-Victor Poncelet over the
period 1819–1839. The former called the quantity
quantité de
travail (quantity of work) and the latter,
travail
mécanique (mechanical work), and both championed its use in
engineering calculation.
In a paper
Über die Natur der Wärme, published in the
Zeitschrift für
Physik in 1837,
Karl
Friedrich Mohr gave one of the earliest general statements of
the doctrine of the conservation of energy in the words: "besides
the 54 known chemical elements there is in the physical world one
agent only, and this is called
Kraft [energy or work]. It
may appear, according to circumstances, as motion, chemical
affinity, cohesion, electricity, light and magnetism; and from any
one of these forms it can be transformed into any of the
others."
A key stage in the development of the modern conservation principle
was the demonstration of the
mechanical equivalent of
heat. The caloric theory maintained that heat could
neither be created nor destroyed but conservation of energy entails
the contrary principle that heat and mechanical work are
interchangeable.
The mechanical equivalence principle was first stated in its modern
form by the German surgeon
Julius Robert von Mayer.
Mayer
reached his conclusion on a voyage to the Dutch East
Indies, where he found that his patients' blood was a deeper red because they
were consuming less oxygen, and therefore
less energy, to maintain their body temperature in the hotter
climate. He had discovered that
heat and
mechanical work were both forms of
energy, and later, after improving his knowledge of physics, he
calculated a quantitative relationship between them.
Joule's apparatus for measuring the
mechanical equivalent of heat.
A descending weight attached to a string causes a paddle
immersed in water to rotate.
Meanwhile, in 1843
James Prescott
Joule independently discovered the mechanical equivalent in a
series of experiments. In the most famous, now called the "Joule
apparatus", a descending weight attached to a string caused a
paddle immersed in water to rotate. He showed that the
gravitational
potential energy lost
by the weight in descending was equal to the thermal energy
(
heat) gained by the water by
friction with the paddle.
Over the period 1840–1843, similar work was carried out by engineer
Ludwig A. Colding though it was little known outside
his native Denmark.
Both Joule's and Mayer's work suffered from resistance and neglect
but it was Joule's that, perhaps unjustly, eventually drew the
wider recognition.
- For the dispute between Joule and Mayer over priority, see
Mechanical
equivalent of heat: Priority
In 1844,
William Robert Grove
postulated a relationship between mechanics, heat,
light,
electricity and
magnetism by treating them all as
manifestations of a single "force" (
energy in modern
terms). Grove published his theories in his book
The
Correlation of Physical Forces. In 1847, drawing on the
earlier work of Joule,
Sadi Carnot and
Émile Clapeyron,
Hermann von Helmholtz arrived at
conclusions similar to Grove's and published his theories in his
book
Über die Erhaltung der Kraft (
On the Conservation
of Force, 1847). The general modern acceptance of the
principle stems from this publication.
In 1877,
Peter Guthrie Tait
claimed that the principle originated with Sir Isaac Newton, based
on a creative reading of propositions 40 and 41 of the
Philosophiae
Naturalis Principia Mathematica. This is now generally
regarded as nothing more than an example of
Whig history.
The first law of thermodynamics
Entropy is a
function of a quantity of heat which shows the possibility of
conversion of that heat into work.
For a thermodynamic system with a fixed number of particles, the
first law of thermodynamics may be stated as:
- \delta Q = \mathrm{d}U + \delta W\,, or equivalently,
\mathrm{d}U = \delta Q - \delta W\,,
where \delta Q is the amount of energy added to the system by a
heating process, \delta W is the amount of energy lost by the
system due to work done by the system on its surroundings and
\mathrm{d}U is the change in the internal energy of the
system.
The δ's before the heat and work terms are used to indicate that
they describe an increment of energy which is to be interpreted
somewhat differently than the \mathrm{d}U increment of internal
energy. Work and heat are
processes which add or subtract
energy, while the internal energy U is a particular
form
of energy associated with the system. Thus the term "heat energy"
for \delta Q means "that amount of energy added as the result of
heating" rather than referring to a particular form of energy.
Likewise, the term "work energy" for \delta W means "that amount of
energy lost as the result of work". The most significant result of
this distinction is the fact that one can clearly state the amount
of internal energy possessed by a thermodynamic system, but one
cannot tell how much energy has flowed into or out of the system as
a result of its being heated or cooled, nor as the result of work
being performed on or by the system. In simple terms, this means
that energy cannot be created or destroyed, only converted from one
form to another.
For a simple compressible system, the work performed by the system
may be written
- \delta W = P\,\mathrm{d}V,
where P is the
pressure and dV is a small
change in the
volume of the system, each of
which are system variables. The heat energy may be written
- \delta Q = T\,\mathrm{d}S,
where T is the
temperature and
\mathrm{d}S is a small change in the
entropy
of the system. Temperature and entropy are also system
variables.
Mechanics
In mechanics, conservation of energy is usually stated as
- E=T+V,
where T is kinetic and V potential energy.
Actually this is the particular case of the more general
conservation law
- \sum_{i=1}^N p_i \dot{q}_i - L=const and p_i=\frac{\partial
L}{\partial \dot{q}_i}
where
L is the Lagrangian function. For this particular
form to be valid, the following must be true:
- The system is scleronomous (neither
kinetic nor potential energy are explicit functions of time)
- The kinetic energy is a quadratic form with regard to
velocities.
- The potential energy doesn't depend on velocities.
Noether's theorem
The conservation of energy is a common feature in many physical
theories. From a mathematical point of view it is understood as a
consequence of
Noether's theorem,
which states every symmetry of a physical theory has an associated
conserved quantity; if the theory's symmetry is time invariance
then the conserved quantity is called "energy". The energy
conservation law is a consequence of the shift
symmetry of
time;
energy conservation is implied by the empirical fact that the
laws of physics do not change with time
itself. Philosophically this can be stated as "nothing depends on
time per se".In other words, if the theory is invariant under the
continuous symmetry of
time translation then its energy (which is
canonical conjugate quantity to time) is
conserved. Conversely, theories which are not invariant under
shifts in time (for example, systems with time dependent potential
energy) do not exhibit conservation of energy – unless we consider
them to exchange energy with another, external system so that the
theory of the enlarged system becomes time invariant again. Since
any time-varying theory can be embedded within a time-invariant
meta-theory energy conservation can always be recovered by a
suitable re-definition of what energy is. Thus conservation of
energy for finite systems is valid in all modern physical theories,
such as special and general relativity and quantum theory
(including
QED).
Relativity
With the discovery of
special
relativity by
Albert Einstein,
energy was proposed to be one component of an
energy-momentum 4-vector. Each of the four
components (one of energy and three of momentum) of this vector is
separately conserved across time, in any closed system, as seen
from any given
inertial
reference frame. Also conserved is the vector length (
Minkowski norm), which is the
rest mass for single particles, and the
invariant mass for systems of particles
(where momenta and energy are separately summed before the length
is calculated—see the article on
invariant mass).
The relativistic energy of a single
massive
particle contains a term related to its
rest
mass in addition to its kinetic energy of motion. In the limit
of zero kinetic energy (or equivalently in the
rest frame) of a massive particle); or else in
the
center of momentum
frame for objects or systems which retain kinetic energy, the
total energy of particle or object (including internal kinetic
energy in systems) is related to its
rest
mass or its
invariant mass via
the famous equation E=mc^2. Thus, the rule of
conservation of energy in
special relativity was shown to be a special case of a more
general rule, alternatively called the
conservation of mass
and energy,
the conservation of
mass-energy,
the conservation of
energy-momentum,
the conservation of invariant
mass or now usually just referred to as
conservation of energy.
In
general relativity
conservation of energy-momentum is expressed with the aid of a
stress-energy-momentum
pseudotensor.
Quantum theory
In
quantum mechanics, energy is
defined as proportional to the
time
derivative of the
wave function.
Lack of
commutation of the time
derivative operator with the time operator itself mathematically
results in an
uncertainty
principle for time and energy: the longer the period of time,
the more precisely energy can be defined (energy and time become a
conjugate
Fourier pair).
However, there is a deep contradiction between quantum theory's
historical estimate of the vacuum energy density in the universe
and the vacuum energy predicted by the
cosmological constant. The estimated
energy density difference is of the order of 10
^{120}
times. The consensus is developing that the quantum mechanical
derived
zero-point field energy
density does not conserve the total energy of the universe, and
does not comply with our understanding of the expansion of the
universe. Intense effort is going on behind the scenes in physics
to resolve this dilemma and to bring it into compliance with an
expanding universe.
Notes
- Lavoisier, A.L. & Laplace, P.S. (1780) "Memoir on Heat",
Académie Royale des Sciences pp4-355
- von Mayer, J.R. (1842) "Remarks on the forces of inorganic
nature" in Annalen der Chemie und Pharmacie,
43, 233
See also
References
Modern accounts
- Goldstein, Martin, and Inge F., 1993. The Refrigerator and
the Universe. Harvard Univ. Press. A gentle introduction.
- Stenger, Victor J. (2000). Timeless Reality.
Prometheus Books. Especially chpt. 12. Nontechnical.
History of ideas
- Kuhn, T.S. (1957) “Energy
conservation as an example of simultaneous discovery”, in M.
Clagett (ed.) Critical Problems in the History of Science
pp.321–56
- , Chapter 8, "Energy and Thermo-dynamics"
External links