PHYS3410
BIOPHYSICS
II
Lecture Notes
Section
I:
Thermodynamics of Biological Systems
Lecture 1: Basic concepts
Suggested
textbooks:
Dilip
Kondepudi and Ilya Prigogine: “Modern Thermodynamics: from
heat engines to dissipative structures.” John
Wiley & Sons, 1998
HGL Coster:
“Thermodynamics of Life Processes” NSW
University Press Ltd. 1981
Historical
aspects: from technology to cosmology
Nearly
2000 years ago Hero of Alexandria made a sphere spin with
a force of steam, but the machines of 18th century
were driven by wind, water and animals. The steam engine revealed
new possibilities: conversion of heat into mechanical motion
(Work = Force acting over distance). The invention of heat
engine started the Industrial Revolution and gave birth to
new branch of physics: Thermodynamics.
With time
thermodynamics evolved from being concerned just with better
efficiency of heat engines to theory that describes transformation
of matter in general. The theory can be applied to a huge
range of systems: one atom to whole universe.
Basic
concepts
Thermodynamic
systems: (i) isolated, (ii) closed and (iii) open
(i)
no exchange of energy nor matter with the exterior
(ii)
exchange of heat and mechanical energy, but not matter
(iii) exchange
both matter and energy
State
of the system specified by macroscopic state variables:
volume V, pressure p, temperature T, mole numbers of the
chemical constituents Nk.
Extensive
variables: proportional to the size of the system (V,
Nk)
Intensive
variables: independent of the size of the system, specify
local property (T, p)
Equilibrium
and Nonequilibrium Systems
If temperature
is not uniform in the isolated system, heat will flow until
state of uniform temperature is established: thermal equilibrium.
The state
of the system specified by p, T, chemical composition, evolves
irreversibly towards a time-invariant state in which there
is no further chemical or physical change in the system: thermodynamic
equilibrium (attractor state).
Nonequilibrium
state can be characterised as a state in which irreversible
processes drive the system to the state of equilibrium, where
these processes will vanish.
Two or
more systems may interact together, exchange energy and/or
matter until they reach thermal equilibrium. If system A is
in equilibrium with system B and this system is in equilibrium
with system C, then A is in equilibrium with C. (Zeroth
Law).
Temperature,
Heat and Quantitative Laws of Gases
17th
and 18th centuries saw fundamental change in man's
conception of nature: scientific approach rather than God's
will. Nature obeys simple universal laws, laws that man can
express in the precise language of mathematics.
Experimentation
and quantitative study started the scientific trend. The thermometer
was constructed at the time of Galileo Galilei (1564 - 1642).
Joseph Black (1728 - 1799) was a professor of medicine and
chemistry at Glasgow. Black drew clear distinction between
temperature (degree of hotness) and quantity of heat. His
experiments established the Zeroth Law. This very
simple law is contrary to experience: metal object feels colder
than a piece of wood at the same temperature….
Black
also discovered specific heat of substances (the amount of
heat needed to increase the temperature of a body depends
on the material, rather than just the mass) and latent heat
of fusion and evaporation of water. Even with these insights
the nature of heat remained an enigma for a long time. Scientists
were misled by believing in a "caloric", which moves
from one substance to another. Only by 19th century
it became clear that heat is a form of energy that could be
translated to other forms.
Temperature
measurement: change of a physical property (eg volume
of a liquid or the pressure of a gas). Uniformity of the unit
depends on the uniformity of the temperature change of the
measured property.
A volume
of gas is a good system to experiment on. Robert
Boyle (1627 - 1691) and Jacques Charles (1746 - 1823)
found the relationship between p, V and T:
pV = NRT (1.1)
universal
gas constant R = 8.31441 JK-1
For many
gases, this law describes experimentally observed behaviour
fairly well for pressures up to a few atmospheres. Johannes
van der Waals (1837 - 1923) proposed an equation in which
he incorporated the effects of attractive forces between molecules.
(p + aN2/V2)(V
- Nb) = NRT (1.2)
a is a
measure of attractive forces between molecules
b is proportional
to the size of molecules
First
Law: Conservation of Energy
The conservation
of the sum of kinetic energy and potential energy is a direct
consequence of Newtons' laws. However, more general approach
was needed for the multitude of thermal, chemical and electrical
phenomena, which were being discovered in 18th
century and later.
Luigi
Galvani (1737 - 1798) and Alessandro Volta (1745 - 1827)
discovered how to generate electricity from chemical reactions
(chemical battery). Michael Faraday (1791 - 1867) demonstrated
how to drive chemical reaction by electricity. The newly discovered
electric current could produce heat and light. Hans Orsted
(1777 – 1851) added generation of magnetic field by electric
current. Thomas Seebeck (1770 – 1831) demonstrated thermoelectric
effect: generation of electricity by heat. Generation of electric
current by changing magnetic fields came in 1831 (Faraday).
This web
of inter-related phenomena represents the transformation of
one indestructible quantity: energy. On a microscopic scale
all energy is ultimately reducible to kinetic and potential
energy of interacting particles. This concept was initially
formulated by James Prescott Joule (1818 – 1889).
Joule
showed that there is an equivalence between heat and mechanical
energy: certain amount of mechanical energy always produces
same amount of heat: 4.184 J produces 1 calorie of heat.
In the
classical picture of particle motion, heat is a disordered
form of kinetic energy.
When a
body is heated or cooled, the average kinetic energy of molecules
changes:
 (1.3)
k = 1.381
x 10-23 JK-1 (Boltzmann constant)
If phase
transformation occurs, heat does not change temperature of
the body, but causes melting or fusion, evaporation or condensation.
The conservation
of energy can be stated in terms of macroscopic variables:
exchange of heat, performance of work and change in chemical
composition.
The algebraic
sum of different energy changes dU: heat exchanged and work
done is independent of the manner of transformation. It only
depends on the initial and final states. So, in time interval
dt, the energy U changes by dU:
dU = dQ
+ dW + dUmatter (1.4)
Q is heat,
W is work, and signs depend on heat transfer from or to the
system and work done on or by the system.
For a
closed system dUmatter = 0
So, for
a cyclic process:
U = U(T,
V, Nk) + Uo (1.6)
Change
from O to X is specified by X (if U at O is arbitrarily defined
as Uo)
The quantities
dQ and dW are not independent of the manner of transformation:
cannot be specified by initial and final states.
Most physics
textbooks do not include irreversible processes (more about
these in next lecture), and all transformations take place
very slowly. Therefore, dQ cannot be defined in terms of interval
dt. The usual strategy is to use imperfect differential dQ,
which depends on initial and final states and the manner of
transformation.
We will
not avoid irreversible processes and will not use imperfect
differentials.
So, dQ
and dW can be specified in terms of the rate laws for heat
transfer and the forces that do the work.
For example:
Q supplied in time dt by a heating coil of resistance R carrying
current I is given:
dQ = (IR2)dt
= (VI) dt (1.7)
where
V is the voltage drop across the resistor.
For an
open system, the volume is often not defined as the volume
occupied by a fixed number of moles, but by the boundary of
the system, perhaps a membrane.
Different
types of thermodynamic work:
1)
work done by the system at pressure P due to change
of volume V:
dW
= PdV
(for a
closed system)
2)
electrical work by transferring electric charge dq
into system at electrostatic potential y:
dW
= ydq
3)
for a change of surface area dA with associated surface
tension g
dW = gdA
4)
chemical work when number of moles of species k is
increased by dNk:
dW
= mkdNk
mk
is the chemical potential of the species k:
 (1.8)
In general,
the quantity dW is a sum of different types of work, each
term being a product of an intensive variable and a differential
of an extensive variable.
Tutorial
Question:
A closed
system contains a resistor R. A current of 1 A is passed through
the resistor, which develops potential difference of 100 V
across it. The current flows for 10 minutes.
The systems
internal energy U is unchanged at the end of this time.
How much
work has the system performed?
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