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Chem 4615 Physical Chemistry for the Life Sciences
Chem 4615
Physical Chemistry for the Life Sciences
Diego Troya
Department of Chemistry
Virginia Tech
Lecture notes
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F.0 Units
-We will be using the SI system of units most of the time (but
not always!).
-We will be using 5 out of the 7 Base SI units:
The unit of length is meter (m)
The unit of mass is kilogram (kg)
The unit of time is second (s)
The unit of temperature is kelvin (K)The unit of amount of substance is mole (mol)
-Resource section 2 gives a great review of units.
-Make sure that you know the prefixes of SI and other metric
units Table 3, p 558
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F.1 The constituents of matter
- As we learned in General Chemistry, matter is composed of
atoms.
- Atoms are characterized by their atomic number Z, which
denotes the number of protons.- Isotopes are atoms with the same number of protons, but
different mass number A (they have a different number of
neutrons).
- In neutral atoms, a number of electrons (which carry negative
charge) identical to the number of protons (positively charged)surround the nucleus.
- Ions are formed by the removal or addition of electrons to a
neutral atom.
- Molecules are formed by the aggregation of atoms.
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F.1 The constituents of matter
- In most biological molecules, the bonding between the atoms
is covalent, and emerges by the sharing of electrons among
atoms. We will study this type of bonding in detail in CHEM
4616.
- Other substances in nature (mostly solids) have ionic ormetallic bonding, but they are less relevant here.
- Two molecules may interact via non-bonding interactions (also
called intermolecular interactions). These interactions might
also take place between two parts of a large molecule. When aprotein folds, sections of the protein come in contact with one
another and interact via non-bonding interactions. Hydrogen
bond is the quintessential example of a non-bonding
interaction, but there are others, like dipole-dipole, dispersion,
etc. (More on this in CHEM 4616).
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F.1 The constituents of matter
A recurring theme in large biological polymers is that they are
composed of simple building blocks.
- Aproteinis composed by aminoacids
-Nucleic acidsare composed by bases, sugars, and
phosphate units-Polysaccharidesare complex molecules formed by sugars
-Lipidsare formed by long hydrocarbon units and polar
headgroups
- There are fourlevels of structurethat are commonly
discussed. For the specific case of a protein, the primarystructure is simply the aminoacid sequence. The secondary
structure is the local spatial arrangement of aminoacids. The
tertiary structure is the global 3D shape of the protein. Finally,
the quaternary structure refers to the way that aggregation of
protein subunits takes place in large proteins(e.g.hemoglobin).
C C S
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F.2. Bulk matter: States of matter
In the end, atoms and molecules aggregate to form bulk matter,
which is what we study in CHEM 4615. A difference has to be
made between thestate of matter and thephysical state.
- We will consider three states of matter: gas, liquid and solid.
Macroscopically:A gas fully occupies a container and conforms to its shape
A liquid does not occupy a container but conforms to its shape
A solid neither occupies a container nor conforms to its shape
At themicroscopiclevel, these states of matter can be
distinguished by molecular motion and separation between
particles. Gas molecules are far away from one another and
travel mostly freely. Liquid molecules are close to one another,
and can move past each other with some restriction. Particles
in a solid are also close to one another but cant move pasteach other.
Ch 4615 Ph i l Ch i t f th Lif S i
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F.2 Bulk matter: Physical states
The physical state of a sample refers to the conditions that
sample is in. To define a physical state completely, one needs
to specify:
i) state of matter
ii) amountiii) volume
iv) pressure
v) temperature
Amountmight be specified either by using moles (e.g. 1.5 mol
of hydrogen), or simply mass (e.g. 16.5 g of oxygen). Volume
is also something intuitive. Notice that the SI units of volume
are m3, but liters (L) are more convenient to use. Please, make
sure that you dominate this unit transformation (1000 L= 1 m3).
PressureandTemperaturerequire more attention, and are
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F.2 Bulk matter. Pressure
Pressure Pressure = ForceArea
P= FAThe unit of pressure is the pascal (Pa). 1 Pa= 1 N m2
There are other more popular units of pressure, such as
atmosphere (atm), bar (bar), torr (torr), and millimeters of
mercury (mmHg).1 atm = 101325 Pa =1.01325105 Pa
1 atm = 1.01325 bar
1 atm = 760 torr = 760 mmHg
The atmospheric pressure we feel ( 1 atm) is due to collisionsof gas molecules with our body. Atop the Everest, there is less
gas, so the pressure is lower. Something that will be important
during this course is the concept of mechanical equilibrium
(Fig. F.4) in a cylinder with a movable piston, which occurs
when the gas inside the cylinder exerts the same pressure on
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Hydrostatic pressure
Under water, we are subject tohydrostatic pressure. This is
the same effect as what was used in the original barometer by
Torricelli (www.ems.psu.edu/ nese/f4 2.gif)The equation of hydrostatic pressure in that barometer can be
derived from the definition of pressure:
pressure= force
area =
mass acceleration
area
Multiplying and dividing by the height of the column, and noting
that the acceleration is due to gravity:
pressure=mass gravity height
area height = densitygravityheight
The resulting equation,p= g hdetermines the atmosphericpressure in the barometer, and will be used when we talk about
osmotic pressure.
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F.2. Bulk matter. Temperature
Temperatureis hard to define. In thermodynamics,
temperature helps determine the direction in which energy will
flow when two objects that are not isothermal are brought in
contact. Energy will flow from the body with higher temperature
to the body with lower temperature until the temperaturesbecome equal. At that point we will have reachedthermal
equilibrium. See Fig. F5.
As you have known for some time, there are three main
temperature scales: Kelvin, Celsius and Fahrenheit. Kelvin is
the SI unit of temperature. Here are some useful
transformations:
oK=o C+ 273.15
oF=o C 95+ 32
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F2. Bulk matter. Equations of state
We have said that, aside from the state of matter, V, p, T, and n
are the four quantities that are needed to specify the physical
stateof a system. Well, it turns out that these 4 quantities are
not independent of each other. This gives rise to the equations
of state, which relate the four quantities.
The easiest equation of state is that for an ideal gas (termed
perfect gas in your book), which was established some 200
years ago by combining Boyles, Charless, and Avogadros
laws:
pV= nRT R = 0.08206 atm L mol1K1 = 8.314 J mol1K1
where R is the ideal-gas constant, and can be expressed in
multiple units (please, give attention to TableF2).
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F2. Bulk matter. Equations of state
Something that emerges from the ideal-gas equation of state is
that themolar volume(volume occupied by 1 mol of gas) is
identical for all gases for a given set of temperature and
pressure.
pV= nRT Vn = Vm= RTp
Atstandard ambient temperature and pressure (SATP)
(p=1 bar, T=298.15 K), the molar volume of any gas that
behaves ideally is 24.8 L/mol.
Atstandard temperature and pressure (STP) conditions(p=1 atm, T=273.15 K), the molar volume of any gas that
behaves ideally is 22.4 L/mol.
The SATP and STP conditions will appear quite often during
the rest of the semester.
Thestandard state(Table F.3) will also beimportant.
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y y
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Equations of state for non-perfect gases
-ThepV= nRTis an equation of state only followed by gasesthat behave ideally. There are two main conditions for ideal
behavior:
i) There are no interactions between the gas particles
ii) The size of the gas particles is negligible compared to theseparation between the gas particles.
-These conditions are satisfied by most gases at ambient
temperature and pressures below ca. 10 atm. Deviations from
ideality are studied by thecompressibility factor(this is not in
the book):
Z= pV
nRT =
pVm
RT
-For a gas that behaves ideally, the compressibility factor
should be 1. Z=1.
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y y
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Equations of state for non-perfect gases
Deviations from ideal behavior (not in the book):
If Z>1 the gases are more difficult to compress than ifideal (there are repulsions, the non-negligible volume of
the molecules starts to factor in)
If Z
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Equations of state for non-perfect gases
1.- Corrections to the pressure term (Pideal= (Preal+ an2/V2)):
Pressure depends on both the number of collisions of the gas
molecules with the walls of the container and the momentum of
the particles colliding. If there are many collisions with the walls
of the gas container and the impinging particles have highmomentum, the pressure will be high. Attractive interactions
among gas molecules reduce both the number of collisions and
their momentum, thus reducing the pressure. Both types of
reductions in the pressure are proportional to the number of
molecules present, orn/V(density of the gas). ais a measureof the strength of the attractions.
2.- Corrections to the volume term: Videal= (Vreal nb)Molecules have a finite volume and this should be taken into
account in a proper description. Notice the minus sign:
Vreal>Videal if molecules occupy a non-negligiblevolume.
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F.3 Energy
Force Force = mass acceleration F= m aThe unit of force is the newton (N). One newton is the force
required to push a 1 kg mass to an acceleration of 1 m s2.
1 N= 1 k g m s 2
Work Work = Force distance W= F dThe unit of work is the Joule (J). One joule is the amount of
work that is done when one newton of force is applied to push
an object over a distance of 1 meter. 1 J= 1 kg m2 s2 = 1 Nm
Energy.Energy is the ability to do work, and it has the same units as
work (J).
There are two types of energy, kinetic (due to motion) and
potential (due to position).
Kinetic energyalways isEK= 12 mv2.
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F.3 Energy
Potential energycan come in a variety of ways.
-The potential energy of interaction between two charged
particles in a vacuum separated by a distanceris described by
Coulombs law: Ep= Q1Q240rwhere the Qs are the charges, and
0 is the permittivity of vacuum. Coulombic energy plays a bigrole in a variety of biomolecules, including DNA.
- Another example of potential energy is gravitational energy.
The potential energy of an object above the ground is:Ep= mghwhere m is the mass of the object, g the gravitationalconstant (9.81m s2), andhis the height above ground.
- Total energy is conserved, and is the sum of kinetic and
potential energies: E= EK+ EP
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F.3 Energy. The Boltzmann distribution
-We will learn in CHEM 4616 that counter to what we
experience in our daily life, energy is not continuous, but
discrete. This becomes more noticeable at the microscopic
level.
-Molecules can have various types of energies: translational,rotational, vibrational, and electronic. Since energy is discrete
at the atomic level, molecules can only have given amounts of
energy, called energy states (see Fig. F8).
-At a given temperature, matter has thermal energy, which
allows the system to have various energy states.-The relative population between two energy states N1 andN2at a given temperature is provided by the Boltzmann
distribution:
N2N1 = e
(E2E
1)/RT
or
N2N1 = e
(E2E
1)/k
BT
kB= R
NA
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F3. Kinetic Theory of Gases
Kinetic Theory of Gases
The ideal gas equation of state is a relation between properties
that define the state of a gas (temperature, pressure, volume,
amount).
Although this law is very useful, it does not provide any insight
into the behavior of gases at the atomic level.
The kinetic theory of gases tries to go beyond the
macroscopic description of gases provided by the gas laws by
introducing simple models that attempt to characterize thebehavior of gases at themicroscopiclevel.
For instance, we are going to try to define pressure and
temperature through investigation of the behavior of individual
gas molecules.
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F3. Kinetic Theory of Gases
Thekinetic theory of gasesis based on three main
approximations:
1) The particles of the gas are in random motion
2) There are no interactions between the gas particles until
they collide. In addition, the collisions are elastic (no energy islost to the surroundings)
3) The size of the particles is negligible compared to the
interparticle separation
With these assumptions, we will be able to calculate the
pressure exerted by a gas based solely on the mass andvelocity of the particles making up the gas:
p=2
3
N
V
whereNis the number of gas particles,Vis the volume, and
< 12 mv2 >is the average kinetic energy ofeachgasparticle.
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Derivation of pressure from the Kinetic Theory of Gases
(This derivation is easier than what you have in the book p267))
- Intuitively, we can think that the pressure of a gas is related to
the force exerted by gas molecules on the walls of the container
that keeps the gas (think about a pressure cooker). P= FA- Lets try to calculate the force exerted by a molecule traveling
perpendicularly to a flat wall of a cubic container at speed vx.
force= mass acceleration= massdistance
time2
force=momentum
time- So the force is the change in momentum per unit time.
-The momentum of the gas molecule before a collision with the
wall ismvx, and mvxright after it (the particle is moving only
in the x direction, and the collision is elastic). Thus the total
change in momentum is 2mvx.
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- It follows that:
P= force
area = Nmv2
xl
1
l2 =Nmv2
xV
- This is the pressure exerted byNmolecules traveling in only
one direction at the same speed vx.
- However,molecules travel randomlyin the three directions
of space (x,y,z). In addition, at a given temperature, themolecules can have abroad range of velocities. To keep
things simple, we are going to do the calculations taking into
accountroot-mean-squaredvelocities in each direction
(,,,=v2x,1+v2x,2+v2x,3+
N ).- The average total squared velocity of the gas particles is:
= + +
- If we have many molecules, ==, thus
= 3< v2x> or = 3
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- Now that we have included the fact that the molecules move
randomly in all three directions of space, and that they might
have a broad range of velocities, we can redefine pressure as:
P= Nm
3V
- Notice that we are using a statisticaltreatment. Use ofaverage velocities is only justified if we have many molecules.
- A more common expression of this equation takes into
account the definition of thekinetic energyof a particle:
= 1
2 m
- With this, the formulation of pressure from a microscopic point
of view is:
P= 2
3
N
V
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Temperature and velocity (kinetic energy)
- We have just defined pressure in terms of molecularproperties. Now we are going to define temperature also at the
microscopic level.
- Intuitively, temperature and molecular motion seem related
(hot molecules move faster). To prove this point, lets insert the
microscopic definition of pressure in the ideal-gas equation:
pV= nRT 2
3
N
V V=
N
NART
- If we solve for< EK>
=3
2
R
NAT=
3
2kBT
whereNA is Avogadros number, andkBis the Boltzmann
constant (R/NA= 1.381 1023
J K1
)
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Important observations:
Temperature and kinetic energy are related. Hence, we
commonly refer to the motion of a gas as thermal motion
Different gases at the same temperature have the same
average kinetic energy, regardless of their mass.
Since we know that< EK>= 12 m, we can find the
velocityof a gas at a given temperature:
1
2m=
3
2kBT
= vrms=
3kBT
m =
3RT
M
- Thatvelocity vrmsis called theroot-mean-squarevelocity ofa gas, and it depends on mass.
- The average kinetic energy of a gas does not depend on
mass (i.e. all gases have the same average kinetic energy at
the same temperature), but the rms velocity of the gas
molecules depends on the mass.
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F3. Kinetic Theory of Gases
Maxwell distribution of velocitiesIn the foregoing discussion,
average (more precisely root-mean-squared) velocities have
been considered. However, in a real gas, not all of the
molecules move at the same velocity. For an ideal gas, Maxwell
deduced that the distribution of velocities can be obtained fromthe following probability distribution:
f= F(s)s with F(s) = 4
M
2RT
3/2(s2)
e
Ms2
2RT
wherefis the probability of finding a particle moving at a speed
betweensands+s.The two most important components of the distribution are the
exponential and the (s2) terms.
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F3. Kinetic Theory of Gases
- The exponential term is aGaussianfunction (see page 14),
which has a bell shape. A Gaussian function has a maximum
around a central point and decays at high or low values of the
velocitys. This means that the probability to find particles that
are moving very fast or very slow is small.- The exponential depends on the mass Min such a way that
the largerM is, the faster the function decays. I.e., the larger
the mass, the lower the probability to find particles moving at
high speeds (see Fig. F11)
- The opposite occurs with the dependence on T. SinceT isdividing in the exponent, the largerTis, the smaller the
exponent is, and the slower the decay. At high T, the molecules
will be able to move faster. (See Fig. F10).
- Finally, the (s2) term becomes small at small s values,
introducing asymmetry in the distribution (SeeFigs.F10,F11)
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Extensive and intensive properties
Throughout the semester, you will sometimes hear the termsextensive and intensive properties.
Extensive properties are those that depend on the size of the
system. Examples of extensive properties are: mass, volume,
energy, heat capacity, ...
Intensive properties are those that do not depend on the size of
the system. Example of intensive properties are: temperature,
pressure, density, and all molar quantities (molar energy, molar
heat capacity, ...)It is generally more convenient to work with intensive
properties. For instance, if you want to report how much energy
is relased by combusting glucose, you can either say: 2808 kJ
for a sample that weighs 180.1 grams, or 2808 kJ/mol. Working
with intensive properties avoids having to specify theamount.
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p
Summary of equations
- Force: F= m a(mass x acceleration)- Work: W= F d(force x distance)
- Kinetic energy: EK= 1
2mv2
- Gravitational potential energyEP= mgh
- Pressure: P=F
A(force over area)
- Hydrostatic pressure: P= g h(density x gravity x height)- Equation of state for an ideal gas: pV= nRT
- Compressibility: Z= pVRT
= VmRT
- Van der Waals equation of state for a real gas:
(P+ an2
V2)(V nb) = nRT
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p
Summary of equations
- Pressure from kinetic theory of gases: p=2
3
N
V
- Temperature and kinetic energy of a gas: EK=3
2kBT or
EK= 32
RT (molar quantity)
- Root-mean-square velocity of a gas: vrms=
3RT
M- Maxwell speed distribution:
f= F(s)s with F(s) = 4
M
2RT
3/2(s2)
e
Ms2
2RT
(this last equation will be provided in a test if needed)
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