Energy, the First Law, and Enthalpy
We have agreed that work, potential energy, kinetic energy, and heat are all forms of
energy. Historically, it was not obvious that heat belonged in this list. But beginning
with the experiments of Count Rumford of the Holy Roman Empire, and later the
experiments of Joule, it became clear that heat, too, was just another form (or
manifestation) of energy.
Work, potential energy, kinetic energy, and heat can be transformed into each other.
Work, potential energy, and kinetic energy can be transformed quantitatively into heat
with no restrictions, but there are restrictions on the transformation of heat into work.
These restrictions are the subject of the second law of thermodynamics and will be
discussed later. Suffice it to say that you can convert heat into work - this is what
engines do - but along with that conversion there will always be some heat loss into the
surroundings.
Once we know that work, potential energy, kinetic energy, and heat are all forms of
energy we can state the first law of thermodynamics. In words, the first law declares that
energy is conserved. That is, energy can neither be created nor destroyed. Energy can
be changed into its different forms, but it can't go away and it cannot be created out of
nothing1. There is another word statement of the first law which is of historical interest.
This statement is, "it is impossible to create a perpetual motion machine of the first
kind2." A perpetual motion machine of the first kind is an engine that produces energy
(or work) without consuming resources of any kind.
Before we can cast the first law into mathematical terms we will have to define a new
quantity and establish some conventions. First of all, we have to define the system and
the surroundings. The system is the object of study. It is the sample of material, or the
apparatus, or the machine, or whatever object or collection of objects that you are
interested in at the moment. The surroundings is everything else. One often sees the
statement that system + surroundings = universe. This may sound a little arrogant, but it
does have the nice feature of allowing us to focus our attention on the system of interest
without worrying about the details of what else is out there. Most often we will idealize
the surroundings as an infinite heat bath at some temperature, T, or as a bomb
calorimeter, or as a pressure bath at pressure, p, and so on.
While we are talking about the system and surroundings it would be good to introduce
some other terms. If there is a boundary, or barrier, between the system and the
surroundings which does not allow the system to expand or contract and prohibits all
exchanges of heat and matter between the system and surroundings we say that the
system is isolated. If the boundary permits the passage of energy (heat, work, etc.), but
not material, we say that the system is closed. If the boundary allows the passage of heat,
work, and material, we say the system is open.
Now we can define a quantity, U, called the internal energy. The internal energy is just
what its name says it is, it is the energy contained in the system. Thermodynamics is not
concerned about whether or not there are atoms and molecules, but we can use our
knowledge of atoms and molecules to give us an intuitive understanding of the internal
energy. The internal energy is the sum of all the kinetic and potential energies of all the
atoms and molecules in the system. We will define several more quantities like U, which
have units of energy, but U is the only one of them which will have an easily understood
physical meaning in terms of kinetic and potential energies of particles.
Before we can put the first law into an equation we must agree on some conventions
regarding work, w, and heat, q. We will define w as the work done on the system. Work,
w, can be positive or negative. If w is positive then work was, indeed, done on the
system, which tended to increase the energy of the system. If w is negative then work
was done by the system on the surroundings which tended to decrease the energy of the
system. Likewise, q is defined as heat absorbed by the system. Heat, q, can be positive
or negative. If q is positive then the system did, indeed, absorb heat, which tended to
increase the energy of the system. If q is negative the system gave up heat to the
surroundings, which tended to decrease the energy of the system. One more thing, we
need an expression for the change in internal energy when we carry out some process.
We will use the upper case Greek delta as the symbol for "change in." Thus we will write
(1)
U  U final  U initial .
In words, this equation reads, "the change in the internal energy is equal to the final
internal energy minus the initial internal energy." This equation also tells us that U is a
"state function." That is, the change in U does not depend on how the change was done
(in other words, on the path), but depends only on the initial and final states.
We can now give the first law of thermodynamics in equation form. For a finite change
we write,
(2)
U  q  w .
For a differential change we write the first law in differential form,
(3)
dU  dq  dw .
The w in Equation 2 or the dw in Equation 3 includes all types of work, work done in
expansion and contraction, electrical work, work done in creating new surface area, and
so on. Much of the work that we deal with in thermodynamics will be work done in
expansion and contraction of the system, or pV work. The expression for pV work is,
(4)
dw   pdV .
If we want to include both pV work and other types of work we can write the first law as,
(5)
dU  dq  pdV  dwother .
Let's now confine ourselves to systems where there is only pV work. In this case the first
law can be written,
(6)
dU  dq  pdV .
Suppose we now regard U as a function of T and V. That is, U = U(T,V). Then, for dU
we can write,
(7)
 U 
 U 
dU  
 dT  
 dV .
 T V
 V T
For a process at constant V (dV = 0) Equations 6 and 7 become,
(8)
and
(9)
dU V  dqV
 U 
 dT .
dU V  
 T V
We also know, from our discussion on heat and heat capacity, that the differential heat at
constant volume can also be written as,
(10)
dqV  CV dT
so,
(11)
dU V  CV dT .
Comparing Equations 9 and 11, and recognizing that the change dUV is the same in both
cases, we see that,
(12)
 U 
 .
CV  
 T V
We shall regard Equation 12 as the formal thermodynamic definition of the heat capacity
at constant volume, much better than our previous temporary definition,
(13)
 q 
 .
CV  
 T V
Equation 12 is a better definition of the heat capacity because it is usually more
satisfactory to define thermodynamics quantities in terms of state functions, like U, T, V,
p, and so on, rather than on things like q and w which depend on path.
One other comment, we can integrate Equation 8, at constant volume, to get,
(14)
U V  qV .
In words, for any process at constant volume the heat, q, is the same as the change in the
internal energy, U.
It turns out that V is not the most convenient variable. It is much easier to control the
pressure, p, on a system than it is to control the volume of the system, especially if the
system is a solid or a liquid. What we need is a new function, with units of energy, which
contains all the information that is contained in U but which can be controlled by
controlling the pressure. Such a function can be defined (created) by a Legendre
transformation. There are particular criteria which must be met in making a Legendre
transformation, but in our case here these criteria are met. (A full discussion of the
mathematical properties of Legendre transformations is beyond the scope of this
discussion. There are more details given in the Appendices to Alberty and Silby.) In our
case we will define a new quantity, H, called the enthalpy, which has units of energy, as
follows,
(15)
H  U  pV .
We can show that H is a natural function of p (in the same sense that U is a natural
function of V) as follows,
dH  dU  pdV  Vdp
(16)
 dq  pdV  pdV  Vpd .
 dq  Vdp
One of the great utilities of the enthalpy is that it allows us to use a state function, H, to
describe the heat involved in processes at constant pressure rather than the heat, q, which
is not a state function. To see this, let's go through the same process with dH that we did
with dU above. Let's regard H as a function of T and p (for now). Then we can write,
(17)
 H
 H 
dH  
 dT  
 T  p
 p

 dp .
T
Consider a process at constant pressure (dp = 0). From the last line of Equations 16 we
conclude that.
(18)
dH p  dq p
and from Equation 17 we get,
(19)
 H 
 dT .
dH p  
 T  p
We also know, from our discussion on heat and heat capacity, that the differential heat at
constant pressure can also be written as,
(20)
dq p  C p dT
so,
(21)
dH p  C p dT .
Comparing Equations 19 and 21, and recognizing that the change dHp is the same in both
cases, we see that,
(22)
 H 
 .
Cp 
 T  p
We shall regard Equation 12 as the formal thermodynamic definition of the heat capacity
at constant pressure. Again, this definition is much better than our previous temporary
definition,
(23)
 q 
 .
Cp
 T  p
Since it defines the heat capacity in terms of the state function, H, rather than in terms of
q which is not a state function.
Just as we integrated equation 8, we can integrate Equation 21, at constant pressure, to
get,
(24)
H p  q p .
In words, for any process at constant pressure the heat, q, is the same as the change in
enthalpy, H. This equation contains no approximations. It is valid for all process at
constant pressure. Equation 24 is vastly more useful than its counterpart at constant
volume because we carry out our chemistry at constant pressure much more often than
we do at constant volume.
1. Nuclear energy was unknown to the original formulators of thermodynamics. We
now know that matter can be converted into energy and vice versa. The "energy
equivalent of matter" is given by the famous Einstein formula, E = mc2, where m is
the mass of the matter and c is the velocity of light. Since the velocity of light is very
large, about 3 x 108 m/s, a small amount of mass is equivalent to a very large amount
of energy. Strictly speaking, the statement, "energy is conserved," should be replaced
by the statement, "energy plus the energy equivalent of mass is conserved." That is,
energy + mc2 is conserved. The conversion of mass to energy or energy to mass in
chemical reactions is so small that it is virtually never observed in chemical problems.
So, for chemical thermodynamics, the simpler statement that energy is conserved is
sufficient.
2. The phrase, "perpetual motion machine of the first kind," implies that there are other
kinds of perpetual motion machines. There is a hypothetical perpetual machine of the
second kind which will be important in understanding the second law of
thermodynamics. It is not appropriate to discuss this perpetual motion machine here.
Suffice it to say that one of the statements of the second law in words will be that it is
impossible to make such a machine.