Video summary
The Carnot cycle, traditionally defined by a sequence of reversible isothermal and adiabatic steps involving an ideal gas, serves as a foundational model for understanding heat engines. However, the principles derived from this specific cycle extend far beyond it to encompass a wide variety of other engine types, such as the Rankine cycle used in steam engines and the Otto cycle found in combustion engines. While engineers often focus on the specific mechanical details of these distinct cycles, thermodynamics allows us to generalize the efficiency limits regardless of the working substance or the exact path taken, provided the processes remain reversible. This universality means that the fundamental relationship between heat input, work output, and temperature reservoirs applies to any cyclic process, not just those strictly adhering to the Carnot sequence.
At the heart of this generalization lies the First Law of Thermodynamics and the concept of entropy as a state function. For any complete cycle, whether it is a standard Carnot loop or a more complex engine design, the net change in internal energy is zero because the system returns to its initial state. Consequently, the net work produced by the engine must equal the difference between the heat absorbed from the hot reservoir and the heat rejected to the cold reservoir. By defining efficiency as the ratio of useful work output to the total heat input, we can express this efficiency in terms of heat quantities. When combined with the Clausius theorem, which relates heat transfer to temperature changes for reversible processes, it becomes evident that the ratio of these heats is directly proportional to the ratio of the absolute temperatures of the reservoirs.
This synthesis leads to the Carnot Theorem, a pivotal result stating that the efficiency of any reversible heat engine operating between two specific temperatures depends solely on those temperatures and not on the nature of the working fluid or the specific cycle design. Mathematically, this efficiency is expressed as one minus the ratio of the cold reservoir's temperature to the hot reservoir's temperature. This formula holds true for all reversible engines, proving that no matter whether the engine uses an ideal gas, a real gas, or any other material, the maximum theoretical efficiency is capped by this temperature ratio. The derivation relies fundamentally on the reversibility of the steps and the Second Law of Thermodynamics, establishing a universal limit for energy conversion in heat engines that transcends the specific mechanics of individual designs like the Rankine or Otto cycles.
Read the full video transcript
okay so what we've seen so far about the
carnot cycle is that if we run a carnot
cycle for an ideal gas we get a heat
engine with an efficiency that's one
minus this ratio of the cold temperature
the cold reservoir to the hot
temperature the temperature of the hot
reservoir
that's not just specific to
the carnot cycle for ideal gases as it
turns out
but for lots of other types of heat
engines as well so first let me explain
what i mean by other types of heat
engines
the carnot cycle
is
a sequence of adiabatic and isothermal
expansions
so we do an isothermal reversible and
isothermal expansion adiabatic expansion
isothermal compression adiabatic
compression each one of those steps is
reversible and either isothermal or
adiabatic
the
net work
we get out of that process we can
calculate numerically
we don't have to do just isothermal and
adiabatic steps though we can imagine
just for the sake of argument let's say
we did an isothermal expansion
from here to here
and then somehow we need to get from our
hot temperature to some colder
temperature but we don't have to do an
adiabatic expansion to get to the colder
temperature maybe we just drop straight
down to the cold temperature so on this
pv diagram
that would correspond to our second step
being not an adiabatic expansion it's
not an expansion at all our volume's not
changing it's just a pressure drop in
order to reduce the temperature so
that's an we could call that an
isochoric constant volume
decrease in temperature
and then we could
reversibly and isothermally compress
to here and then do another isochoric
heating
to get back up to the initial conditions
so that's another
heat engine we would end up using heat
to do work in that process
so that's not the carnot cycle it's it's
a different cycle so there's a number of
these different types of
cycles
there's a few that we won't dwell on the
details of the rankine cycle is a
sequence of adiabatic followed by
isobaric
expansions and compressions
that corresponds to what happens in a
steam engine
the auto cycle
is corresponds more to what happens in a
combustion engine those are different
steps it's in fact more than just four
steps involved in an auto cycle
but
engineers chemical engineers care a
great deal about the details of specific
flavors of heat engines chemists
are less interested in the details of
specific types of heat engines and more
interested in the thermodynamics of the
process so we're going to not delve into
the details of these different heat
engines too much
but the question is
how general is this statement what can
we say about heat engines in general
even if we're doing something that's not
a carnot cycle
so let's
go back to again thinking about the
carnot cycle but in a more general way
so
again we had four different steps
for the carnot cycle and if i just
remind you
without as much of the math
we had
at the hot temperature and cold
temperature we had we absorbed heat from
the surroundings during the first
expansion step and we gave some heat
back to
the colder surroundings in the third
step the isothermal compression step so
we had
in step number one
reversible isothermal expansion absorbed
some heat
adiabatic expansion there was no heat
isothermal compression we gave some heat
back to the surroundings so these two
terms have opposite signs from each
other and then
no heat in the final adiabatic expansion
adding those up there was some net
amount of heat
from that process
i'm going to skip all the rest of the
details of this diagram and just remind
you
that
for a cyclic process any cyclic process
whether it's a rankine cycle or an otto
cycle any cyclic process is going to
have zero
net change in the internal energy
because it's a state function
and the work done by this process again
even if it's not a carnot cycle even if
it's some other type of cycle the work
is the area enclosed by this diagram so
there's some total amount of work
so
what this tells us what the first law
tells us is these two amounts of heat
plus that amount of work must add up to
the net change in energy zero
so
q cold
plus q hot plus work is zero
or
if i move the work to the opposite side
of the equal sign negative w
is q
hot plus q cold
that's enough information i don't really
have to delve into the details of this
diagram and what they are for an ideal
gas or van der waals gas or some other
type of gas
if i'm interested in the efficiency
of this process remember that's the
amount of work i get out
compared to the amount of heat i had to
put in
just the first law is enough to tell me
that the amount of work i get out is
equal to
negative w is equal to
qh
plus qc
the net heat for the process divided by
qh
that ratio is the efficiency
qh over qh is one
qc over qh
is qc over qh
so
written in terms of heat the efficiency
of the process is 1 plus this ratio of
the heats and remember
qc is a negative number qh is a positive
number so that ratio that's going to end
up decreasing the efficiency below one
we had seen previously that those this
expression
involves temperatures not heats
so let's remind
ourselves the clausius
theorem
which says remember
a change in entropy an infinitesimally
small change in entropy is is dq for the
reversible uh process divided by t
each one of these steps as we've
described them was reversible reversible
isothermal reversible adiabatic
reversible isothermal reversible
adiabatic so all our steps are
reversible so we can use the clausius
theorem and relate the heats for our
steps to entropies so i can say for this
total process
for my four-step process going around
this full cycle the net change in
entropy
is going to be
heat divided by temperature so qh
divided by temperature and that happens
at the hot temperature
this step had no heat
this step the heat was q cold
and that happened at the colder
temperature tc
and then the fourth step had
no heat
so
total change in entropy is the sum of
these terms of which only two of them
contribute anything
but
entropy is a state function so when i
start here i go all the way around the
cycle
when i get back to where i started the
net change in the entropy has to be 0
for that state function
so
this term plus this term have to add up
to 0
qh over th must be the negative of qc
over tc
or since what i'm interested in is
ratios of q's or ratios of ts let's
rearrange this a little bit and write
q
c over qh
so i'm going to take this qh and bring
it down underneath the qc
and then i have to take that tc and put
it above the th
so qc over qh and there's a negative
sign involved
is negative of tc over th that's what i
get by rearranging this equation so i
can use that the clausius theorem
applied to this cyclic process
tells me that this ratio of heats is
negative the ratio of the temperatures
so if i plug that in here i find that
the efficiency is one
instead of plus qc over qh i've got
minus
tc over th
so that's
not surprising because we've seen it
before we saw it for the case of an
ideal gas for the carnot cycle
what we've just done here notice i
haven't said anything about ideal gases
there were no nrt
one over v's or log v's or anything like
that that came from the ideal gas law
only the first law is essentially the
only fact we had to use here so
at least for the carnot cycle where i
have some adiabatic steps we've seen
that the efficiency is always going to
be one minus this ratio of temperatures
whether it's an ideal gas a real gas any
type of
material
the efficiency can be given by this form
this is in fact called the carnotherum
and it's even more general than just the
carnot cycle all types of heat engines
all cyclic processes
on this pva diagram will have
an efficiency given by
this result even
regardless of what kind of heat engine
they are as long as the processes are
reversible so that's the carnot theorem
the carnot theorem says
any reversible heat engine
has
this efficiency
proving that for something other than
carnot cycle uh is a little more
involved in particular it requires us to
make use of the second law of
thermodynamics which we haven't covered
just yet so i'll postpone that for a
little bit but but it's useful to be
able to use that result now and say
regardless of what type of heat engine
we have the efficiency is given by this
result one minus the ratio of cold
temperature over hot temperature for any
reversible heat engine operating between
two heat reservoirs at these two
different temperatures
so
what we'll do next is now that we
understand how heat engines are good at
converting heat into work
it's also sometimes useful to do the
reverse process converts some work into
heat so that's the next thing we'll talk
about