Energy is Conserved: The First Law of Thermodynamics
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Rationale of the Thermodynamics Game
The First Rule of the Game
Thermodynamics is the study of the patterns of energy change. The
"thermo" refers to energy, and "dynamics" means patterns of change. Look at our
schedule and
notice that roughly 2/3 of this course will be concerned with understanding the
patterns of energy change.
I think the most pleasant way to learn about thermodynamics is to imagine
it's a game. This game is about understanding the patterns of energy change
and how these changes relate to the states of matter. That's the concept
behind the game we will be playing.
More specifically, thermodynamics deals with (a) energy conversion and (b)
the stability of molecules and direction of change. What does this mean? Two
examples may illustrate the ideas. First, imagine a brick resting on a window
ledge 3 stories high. As the brick rests on the ledge, it has potential energy
(mgh). If you knock the brick off the ledge the potential energy is converted to
kinetic energy as the brick accelerates toward the ground. Then when the brick
hits the ground the kinetic energy is converted to light energy (sparks), sound
energy (a bang), and chemical energy (the brick breaks). A second example is
protein folding. Proteins are polymers of amino acids connected by peptide
bonds. Proteins fold into their lowest-energy state for the environment and
conditions the proteins are in. Therefore, if you change the conditions you can
change the structure of the protein; as you heat a properly folded protein from
room temperature on up it will eventually unfold into what's called a random
coil. The energetics determine the structure; therefore, thermodynamics can help
one understand the stability and structure of biomolecules.
Before learning the rules of the thermodynamics game and then
playing it, it might be useful to measure up the playing field. Therefore, it's
necessary to define some terms and concepts that we will be using throughout our
study of thermodynamics.
While studying thermodynamics, we will introduce boundaries into the objects
we are considering. These boundaries in our "playing field" are the
system and surroundings. One thing to remember is that we set up
these boundaries: we define the system and the surroundings in a way most
conducive to understanding the energetics of what we're studying. So if you want
to understand what happens when you heat a pot of water on your stove, the first
thing to do is to define the system and let everything else be the surroundings.
You might say that the pot and the water are the system you're considering; or
the pot, water, and stove; or even the pot, water, stove, and kitchen. Defining
the system and surroundings is arbitrary, but it becomes important when we
consider the exchange of energy between the system and surroundings and
subsequently make judgements about the energetics of this exchange.
Two types of exchange can occur between the system and surroundings: (1)
energy exchange (heat, work, friction, radiation, etc.) and (2) matter exchange
(movement of molecules across the boundary of the system and surroundings). This
makes a lot of sense when you consider that the universe is made up of matter
and energy--of course they are also kind of the same thing (E = mc^2). Based on
the types of exchange which take place or don't take place, we will define three
types of systems:
- isolated systems: no exchange of matter or energy
- closed systems: no exchange of matter but some exchange of energy
- open systems: exchange of both matter and energy
While we
are trying to understand the energetics of some system of interest, it's also
useful to make a distinction between different forms of energy. One distinction
we will make is between heat energy and work energy. Early
scientists and engineers studying thermodynamics wanted to use energy to do
useful work, so it's pretty clear why making the distinction between heat and
work can be important. Heat is the exchange of thermal energy from a hot
body to a cold body. When a hot body and a cold body have contact, heat will
flow from the hot body to the cold body until they both reach thermal
equilibrium (they are at the same temperature). Work involves the net
directed movement of matter from one location to another. Some examples of work
are pressure-volume (PV) work, electrical work, and mechanical work. In this
course, we'll mainly be studying PV work (work involving the expansion and
compression of gases).
Taking a look at mechanical work can help illustrate the utility of sign
convention (when work is done on the system by the surroundings, is that a net
negative energy change or a net positive energy change?). We need to set-up a
frame of reference and agree that we will all use this reference so we will
communicate effectively. The question is: when considering the transfer of
energy, should be use the system or the surroundings as our reference? By
convention, we can imagine that we are in the system so if work is done on the
system by the surroundings that will mean there is positive work (the energy in
the system increased). By the same token, if work is done by the system on the
surroundings that will mean there is negative work (the energy in the system
decreased).
We will make a distinction between thermodynamic processes which (a) happen
slowly and can be reversed, and (b) happen so quickly that they can't be
reversed. We simply call processes then:
- reversible: if the process happens slow enough to be reversed.
- irreversible: if the process cannot be reversed (like most
processes).
Some more definitions are as follows:
- isobaric: process done at constant pressure
- isochoric: process done at constant volume
- isothermal: process done at constant temperature
- adiabatic: process where q=0
- cyclic: process where initial state = final state
As you learned earlier,
energy can be exchanged between the system of interest and its surroundings.
However, the total energy of the system plus the surroundings is
constant. That's the First Law of Thermodynamics. The First Law is also
stated as energy is conserved.
How do we know this? This is an empirical law, which means that we know that
energy is conserved because of many repeated experiments by scientists. It's
been observed that you can't get any more energy out of a system than you put
into it . James Prescott Joule did a famous experiment which demonstrated the
conservation of energy and showed that heat and work were both of the same
nature: energy. His system of interest was water in a thermally insulated
container. In this container was also a paddle which was connected to the
outside world (surroundings) and connected to weights on a string. Joule
measured the work done by the paddle wheel and he also measured the heat created
by the wheel turning in the water. Significantly, Joule found that the amount of
energy done as work was converted exactly to heat. Energy was changed from one
form to another (work to heat); however, no net change of energy in the system
plus the surroundings occured. Energy is conserved.
So we're playing this game
of trying to understand the way energy is exchanged and how that influences the
states of matter. There are a few basic "plays" that everyone needs to learn.
One of them is to break down (or analyze) energy into it's various forms like
work, heat, or radiation. We'll start out by learning about the energetics of
work.
When you do work, you force an object(s) in
space--you move something(s) in a net directed fashion. The important ideas here
are that you exert a force on an object through a certain
distance. You might recall that work is actually defined as the product
of a force times a distance.
work = external force x distance
It's important, again, that we communicate effectively and make sure that we
have the same frame of reference (have the same sign convention). If work is
done on the system and the energy of the system increases then we will say that
the sign of the work is positive. On the other hand, if work is done by the
system and the energy of the system decreases then we will say that the sign of
the work is negative. Looking at some examples will help illustrate the point:
Work done when a spring is compressed or extended. Imagine you have a
spring in your hands. In order to compress the spring or extend the spring, you
will have to apply a force. You will have to push the two ends together or pull
them apart. Hooke's law tells us that the force applied is directly proportional
to the change in length of the spring. You can appreciate this intuitively
because you can feel that the more you push against the ends of your spring and
compress it, the harder it will get to compress it and you will have to exert
more and more force. Different springs have different tolerances to the
compressive and extensive forces; therefore, one needs to introduce a measure of
this resistance--the spring constant--in order to equivicate force and
displacement of a spring. According to Hooke's law,
spring force = - k (x - x0),
where k is the spring constant, x0 is the equilibrium position, and x is the
final position. The negative sign shows that the direction of the spring force
is opposite the direction of the displacement from x0. The external force is
equal in magnitude but opposite in sign to the spring force, so
external force (the force of your hands) = k (x -x0).
Now, we want to calculate the work done when we stretch the spring from
postition 1 to position 2. As mentioned before, the force changes as a function
of distance so to calculate the work (which is force times displacement) we will
have to integrate the force with respect to infinitely small changes in
distance:
work, W = † F dx = † k (x - x0) d(x-x0) = 1/2 k [(x2-x0)^2 - (x1-x0)^2] = k
(x2-x1) ((x2-x1)/2 - x0)
Work done when a volume is increased or decreased. Now we will
consider a gas in a container with a movable piston on top. If the gas expands,
the piston moves out and work is done by the system on the surroundings.
Alternatively, if the gas inside contracts, the piston moves in and work is done
by the surroundings on the system. Why would the gas inside contract or expand?
It would if the external pressure, Pex, and the internal pressure, Pin, were
different. To calculate the work done in moving the piston, we know that the
force = pressure times area and then work equals pressure times area times
distance or work equals pressure times the change in volume. So, W = the
integral of (-Pex) dV
Remember, there are two types of PV work:
- Expansion: where Pin > Pex and V2 > V1; work is done by the system
and, therefore, w < 0 and E < 0
- Compression: where Pin < Pex and V2 < V1; work is done by the system
and, therefore, w > 0 and E > 0
In many cases, a PV work
experiment is done in an open system with the Pex a constant value--the
atmospheric pressure (about 1 atm). In this case, W = (Pex)(V2-V1).
Heat (q), like work, is a
form of energy. Heat energy moves from a hotter body to a colder body upon
contact of the two bodies. If two bodies at different temperatures are allowed
to remain in contact, the system of two bodies will eventually reach a thermal
equilibrium (they will have the same temperature).
Different bodies can have different capacities to give up or take in heat.
This capacity of an object to give up or take in heat is called the heat
capacity, C. A more precise definition of heat capacity is the amount of
heat required to raise an object's temperature by a certain amount of degrees.
An even more precise definition is the amount of heat required to raise the
temperature of a substance by 1 degree K. This last definition is what we'll use
in thermodynamics; it can be stated mathematically as
C = (dq)/(dT)
Multiplying both sides of this definition by dT and then integrating both
sides give the following equation (in terms of heat):
q = the integral of (C) dT
The heat capacity of an object depends on its mass. In order to compare two
objects' ability to take in or give up heat (heat capacity), it is useful to
compare the objects on equal terms; therefore, we will define the molar heat
capacity (C "bar") as the heat capacity divided by moles. It can also be
useful to express the heat capacity as the specific heat capacity (C*),
which is the heat capacity divided by the kg of the substance.
Matter can exist in various
states (having a certain density, color, heat capacity, phase, etc.) . Given the
values of T, P, V, and n of a sample of a pure substance, we will know it's
state. Moreover, we know that whenever the matter is in that state it
will have the same properties. Early experiments on the variables of
state (such as T, P, V, and n) showed that only two of these variables of
state need to be known to know the state of a sample of matter. Once two
variables are known, the state of the matter is known and the values of the
other variables can be determined.
The variables of state can be divided into two types--extensive
variables and intensive variables.
- Extensive variables: depend on the amount of substance present.
Examples include the volume, energy, enthalpy, and heat capacity.
- Intensive variables: do not depend on the amount of substance
present. Examples include the temperature and pressure.
One thing to
note is that any extensive variable can be converted to an intensive variable by
dividing it by the moles or the mass (like we did with the heat capacity).
An equation of state is
an equation which relates the variables of state (T, P, V, and n). It's
particularly useful when you want to know the effect of a change in one of the
variables of state. Let's look at some situations where the variables of state
change:
- Solids and Liquids: If the pressure on a solid or liquid is
increased, the volume does not change much. If the temperature is increased,
the volume doesn't change much either. Therefore, an appropriate equation of
state describing such systems would be: V(T,P) = constant.
- Gases: In contrast, changing the pressure or temperature of a gas
will have an easily observable effect on the volume of that gas. For an
ideal gas (no intermolecular interactions and no molecular volume), an
appropriate equation of state would be: V(T,P,n) = (nRT)/P. There are many
equations of state describing real gases. These equations take into
consideration molecular volume and interactions. The most well-known such
equations is probably the Van der Waals equation: [P + (an^2)/V^2)] [V - bn] =
nRT, where a is an experimentally determined constant for molecular
attractions and b is a volume correction for the size of the gas molecules.
Now that we have covered the
equations which define the state of a system, it will be interesting to extend
this work and take a look at the effect of changes of state. We start by
defining the internal energy, E, and the enthalpy, H. Then, we'll consider
changes in heat and work, temperature and pressure. Finally, we will study both
physical and chemical changes.
The internal energy,
E, is just the energy of the system. In thermodynamics, it's useful to define
the energy in another way, as the energy plus pressure times volume. This new
definition is the enthalpy, H = E + PV. For PV work at constant pressure, the
work done is -PV, so you can see that the +PV term in the definition of enthalpy
is a correction for the work term in the energy.
We are strictly interested in the changes between the initial and final
states of energy and enthalpy because energy and enthalpy are variables of state
and depend only on the state of the system. They do not depend on the path used
during the change of state. Moreover, you don't need to be concerned with the
absolute energy of the two states you're studying; instead it's the difference
in the energy between the two states that will be of primary interest. Let's now
look at calculating the energy and enthalpy changes that occur when a system
changes state, and let's consider calculating the amount of energy needed to
cause a desired change in the state of a system.
The energy of a system
will change if heat is transferred to or from the system or work is done by the
system. If heat and work are the only forms of energy transferred between the
systems and surroundings of a closed system, E2 - E1 = q + w. That's the first
law.
- Gas expansion at constant temperature. The isothermal expansion of
a gas can proceed by two types of paths: reversible and irreversible.
- Reversible path: In this case, the changes in pressure at any time are
very small and the direction of the volume change can be reversed. Pex = Pin
- dP.
- Irreversible path: The direction of the volume change cannot be
reversed, and the external and internal pressures will try to reach an
equilibrium. The work done, w = - † Pex dV.
- Cyclic paths. In a cyclic path, the initial and final states are
the same. Therefore, the change in energy and the change in enthalpy are both
equal to zero. All changes in thermodynamic variables of state are equal to
zero.
The
Temperature Dependence of E:
- Liquids and solids at constant P. q = (the integral of) Cp dT = Cp(T2-T1);
w = 0. E = q + w = Cp (T2-T1).
- Liquids and solids at constant V. q = (the integral of) Cv dT = Cv(T2-T1);
w = 0. E = q + w = Cv (T2-T1)
The Temperature Dependence of
H:
H = H2 - H1 = E + P2V2 - P1V1
Temperature and Pressure Changes
for a Gas:
- constant P: q = Cp(T2-T1); w = -Pex(V2-V1)
- constant V: q = Cv(T2-T1); w = 0
- constant T: w = - † (Pex dV) = † (Pin dV) since it's reversible.
Substituting PV = nRT gives w = - † (nRT/V dV) = -nRT ln(V2/V1); for an ideal
gas the change in E is a function of temperature only and E = q + w = 0, so
-q = w = nRT ln(P2/P1)
The following are
various types of phase changes that a sample may undergo:
- fusion (melting) = solid to liquid
- freezing = liquid to solid
- vaporization = liquid to gas
- condensation = gas to liquid
- sublimation = solid to gas
All of these phase changes will either
consume or liberate energy based on the energy associated with intermolecular
interactions--like van der Waals forces, electrostatic interactions, dipolar
interactions. In addition, energy is required for the PV work involved in going
from a liquid to a gas, for example.
Most experiments are done at constant pressure (open to the atmosphere), and
therefore the change in enthalpy will be = q. We will then consider the heats of
fusion, freezing, vaporization, condensation, and sublimation.
The change in energy due to a change in phase is E = H - P V.
E and H for
chemical reactions
At constant P, q = H. For an endothermic
reaction q < 0; for an exothermic reaction q > 0. A bomb calorimeter is a
closed reaction vessel (the volume is constant). In this case, there's no PV
work and E = H. q = E = H = Cv(T2 - T1).
Hess'
Law
The H values for many elementary processes (like 2 H2 + O2
--> 2 H2O) have been determined. This becomes very useful in thermodynamics
since Hess' Law states that we can calculate the overall H for a chemical
rection by simply adding up the H's of the elementary reactions that make up
the overall reaction. So if we want to know the H for A-->B, but only
know the H values for A-->I1-->I2-->I3-->B, we calculate the
overall H as H1 + H2 + H3 + H4 (where the H's correspond to each
step in the chain of elementary processes).
Bond Energies
(Enthalpies)
Energy is released when a bond is broken and is consumed
when a bond is made (there is energy contained in covalent bonds). The bond
energies for many atom pairs have been experimentally determined and can be
found in tables. Using bond dissociation data, we can calculate the energy
required to form a molecule from its constituent atoms (the enthalpy of
formation, Hf) or to dissocate a molecule into smaller groups of molecules or
atoms.
For the decomposition reaction, 2 H2O2 --> 2 H2O + O2, the molecules are 2
(H-O-O-H) --> 2 (H-O-H) + O=O. For the reactants, there are 4 O-H bonds and 2
O-O bonds. For the products, there are also 4 O-H bonds but only 1 O=O bond. The
overall H of the reaction = H(O=O) - 2 H(O-O). The H to form O=O is
-498.3 kJ/mol and the H to form O-O is -143 kJ/mol (x2=-286 kJ/mol). The
overall H of the reaction is then -498.3 + 286 = -212.3 kJ/mol.
Updated: 9/9/95
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