BASICS OF THERMODYNAMICS OF LIVING
SYSTEMS
Thermodynamics deals with mutual conversion of
different types of energy, the direction of physical
and chemical processes and of equilibria. It also
studies systems composed of many parts.
As a
system
we consider any region of space separated from its
surroundings
According to the interaction of the system with its
surroundings we discriminate systems:
isolated – do not exchange matter or energy with
surroundings
closed – exchange only energy with surroundings,
not matter
open – exchange both matter and energy with
surroundings
Thermodynamics studies two types of parameters:
extensive parameters – characterize thermodynamic
system as a whole (mass,
volume, total electric charge)
intensive parameters – they have different values
in different parts of the
system (concentration of chemical components,
temperature, electrical potential)
The studies of the relationship between extensive and intensive
parameters create the basis for the formulation of thermodynamic
laws.
The basic laws of thermodynamics are:
law of conservation of mass
I. law of thermodynamics
II. law of thermodynamics
III. law of thermodynamics
I. law of thermodynamics
If a system is doing a work or the surroundings is doing
a work on the system, its internal state is changed.
E.g. if we compress a gas in a cylinder with a piston
the temperature of the gas increases. Similarly, if there is
a chemical reaction between the components of the
system, its temperature changes. Or, if you consider an
iceberg moving on rocky surface, the friction produces
heat and the iceberg changes its phase – it melts. The
cause producing the change of the state is called energy.
Energy can be thus defined as the ability to change given
(equilibrium) state of matter.
Initial experimets indicated an equivalence between heat
and mechanical work (the work produces heat and heat
can be used to do a work)
This studies led to the formulation of
principle of energy conservation.
This principle can be formulated in different ways, e.g.:
It is not possible to construct a machine generating energy
from nothing. That means it is not possible to produce a
perpetuum mobile of the first kind.
In a more general formulation:
The total energy of isolated system is constant during all
processes.
So we can expres the I. law of thermodynamics in this way
The total energy that a system exchanges with
surroundings in any process is dependent only on
the initial and final state of the system, and not on
the way this change was achieved.
This means there is an energetic function, whose
difference between initial and final state corresponds to
energy exchanged between the system and surroundings.
This function is called
inner energy of the system and is labelled as U.
ΔU = U2 - U1 = q - w
here q indicates heat accepted by the system from surroundings, w
is a work done by the system, indexes 1 a 2 indicate initial and
final state of the system
The I. law of thermodynamics implies that total heat
released in a chemical reaction will be the same if the
reaction proceeds in one step or in more steps.
E.g. the amount of heat released during reaction:
C + O2 = CO2
equals the sum of heat produced in the following reactions:
C + 1/2O2 = CO
CO + 1/2O2 = CO2
This conclusion is known as the Hess law.
Now we can introduce new thermodynamic function.
It is called enthalpy, labeled H, and defined by an
equation:
H = U + PV
where P is pressure and V volume of the system
Now we can calculate the amount of heat released in
the system under constant pressure:
qP = H2 - H1 = ΔH
This expression says that the change of enthalpy in
any process is dependent only on the initial and final
state of the system. In the case of chemical reaction it
is the state of the reactants at the beginning of the
reaction and the state of products in the end of the
reaction.
Reaction heat is the amount of heat exchanged by the
system with surroundings during the chemical reaction.
If the heat is released we speak of an exothermic process,
if the heat is consumed by the system, it is referred to as
endothermic process.
If the reaction proceeds under constant volume, the reaction
heat corresponds to the change of inner energy of the system.
If the reaction proceeds under constant pressure, the reaction
heat corresponds to the change of enthalpy.
II. law of thermodynamics
By the beginning of 19th century Carnot studied the
efficiency of heat machines. He created a concept of
cyclically working heat machine, in which the volume in
the cylinder was changed by interaction with two heat
exchangers having different temperature. Theoretical
work out of this concept led to the formulation of the
theorem:
All the reversible machines working between the same heat
exchangers have the same efficiency in spite of the
composition of the exchangers.
Related formulation was stated by Clausius:
It is not possible to construct an equipment that would do
nothing else than transfer heat from the colder body to a
warmer body.
This implies that it is not possible to create the so called
perpetuum mobile of the second kind.
These formulations are the expressions of the
II. law of thermodynamics
The studies of the efficiency of heat engines revealed
the existence of a new state function called
entropy labeled S
dS = dq/T
According to Carnot theorem the efficiency of reversible
machine is maximum. Thus, the irreversible machines
have always lower efficiency. For the irreversible process
we get:
dS > dq /T
If the system does not exchange heat with surroundings
we get for irreversible process:
dS > 0
and for reversible process:
dS = 0
It means that entropy is growing under irreversible
processes and in equilibrium, when only reversible
processes can proceed, it does not change. Entropy can
be looked upon as a measure of spontaneousness, as it
increases during spontaneous processes.
Energy functions F and G
For the case of reversible process we get from the
I. law of thermodynamics:
dU = dqrev - dwrev
For dqrev we substitute from the definition of entropyTdS:
dU = TdS - dwrev
From this equation we can deduce that work done by the
system under reversible conditions can be expressed using
basic thermodynamic parameters T, U, S.
We can substitute for TdS :
TdS = d(TS) - SdT
In the case of a process under constant temperature
dT = 0 and TdS = d(TS), and the equation can be rewritten
dU - d(TS) = d(U - TS) = -dwrev
For a finite change we get:
Δ(U - TS) = -dwrev
It is evident there is a state function (U - TS),
the decrease of which indicates maximum (i.e. reversible)
work that the system can do under constant temperature.
It is known as Helmholtz function and labelled F:
F = U - TS
We can discern between volume work wvol (wobj = PdV)
and an useful work w,rev, comprising all other kinds of
work (electrical, transport, etc.)
Then we can rewrite the equation:
dU = TdS - PdV - dw,rev
In the case that in the system proceeds reversible process
under constant pressure and temperature, we have:
dT = 0, TdS = d(TS)
dP = 0, PdV = d(PV)
Substituting in the above equation we get:
dU - d(TS) + d(PV) = d(U - TS + PV) = -dw,rev
And for the finite change:
Δ(U - TS + PV) = -w,rev
We can see another state function (U - TS + PV), the
decrease of which indicates maximum useful work that
can be done by the system under constant pressure and
temperature. It is called Gibbs function and labeled G:
G = U + PV - TS = H - TS
III. law of thermodynamics
The formulation was developing in time. As a definitive
version is considered the formulation by Planck from
1912:
Entropy of every chemically homogenous condensed phase
approaches with decreasing temperature zero.
Another formulation explains it more clearly:
It is not possible to cool a physical body to absolute zero in
a finite number of steps.
Changes of entropy in living systems
For the description of internal processes in the system we
consider the states of the system as a whole.
Equilibrium state is reached by a system that is isolated
from surroundings and let suficient time to evolve until it is
not changing any more. This final state will correspond to
the most probable arrangement, characterized by the
highest degree of disorganization, when entropy reaches
its maximum value.
Chemical reactions are characterized by equilibrium
constant K, which describes the composition of the
reaction mixture under situation when the reaction
rate from left to right equals the rate from right to left.
For the change of Gibbs function in equilibrium state
we obtain:
- ΔG = RT ln K.
Living systems are open systems. In the living biological
system taken as a whole we can not expect thermodynamic
equilibrium, as the system in equilibrium can not do work.
However, the ability to do work is essential for the
maintenance of living functions. Open systems are able to
generate certain stationary state, under which the
parameters of the system preserve constant levels of
exchange of matter and energy with surroundings.
The total entropy of an open system can be changed
either due to exchange with external surroundings deS,
or due to internal processes in the system diS:
dS = deS + diS
For the rate of entropy change we obtain:
dS/dt = deS/dt + diS/dt
deS/dt corresponds to the exchange of entropy between
the system and surroundings and it can reach both
positive and negative values,
diS/dt is only positive.
Under stationary state the rate of entropy production is
constant, thus
dS/dt = 0, and therefore│deS/dt│ = diS/dt
dS/dt = deS/dt + diS/dt = 0
If we rewrite this equation:
dS/dt + (-deS/dt) = diS/dt
We can express it in words:
Under stationary state the sum of the rate of entropy
production in the system and the rate of emerging entropy
from the system equals the rate of entropy production
inside the system.
Development and growth of organisms is accompanied by
an increase in the complexity of their organization. From
the point of view of classical thermodynamics it appears
as spontaneous decreasing of entropy of living systems,
which is evidently in contradiction with II. law of
thermodynamics.
However, the decrease of the total entropy of living
organisms appears under conditions of deS/dt < 0
and │deS/dt│ > diS/dt.
It means that the decrease in entropy inside the living
system runs at the expense of increased entropy in the
surroundings.
Let us consider open system in equilibrium under
constant temperature and pressure with no irreversible
processes running, such as heat transfer etc. In such a
system entropy increases only as the result of chemical
reactions, mass transfer between phases of the system
and generally in processes characterized by a change of
chemical potential. We shall consider the heat exchange
to proceed only by reversible processes and then we get
for entropy change:
dS = dqrev/T + diS
We can calculate the entropy production inside the system
diS = dS - dqrev /T
and after aritmetical rearrangement we get:
diS = - dG/T
That can be expressed in words:
The increase in entropy of open system due to internal
nonequilibrium processes is proportional to the decrease
of the Gibbs function of the system.
If we will study the changes of the state parameters in
chemical reactions, when changes in the number of moles
appear, we obtain expressions for the chemical potential μ:
(dU/dni)S,V
(dH/dni)S,P
(dG/dni)T,P
(dF/dni)T,V
=
=
=
=
(μi)S,V
(μi)S,P
(μi)T,P
(μi)T,V
Concomitantly:
(μi)S,V = (μi)S,P = (μi)T,P = (μi)T,V = μi
The relationship between entropy increase, decrease of
Gibbs energy, change in composition and in chemical
potentials in open system can be expressed like this:
diS = - dG/T = - 1/T Σ μidni
diS/dt = - 1/T (dG/dt) = - 1/T Σ μi (dni/dt)
dni = υi dλ
and
dλ = 1/ υi dni
The reaction rate is given:
v = 1/ υi (dni/dt)
After substituting for dni we get:
v = dλ/dt
For the change of mols of reacting substance we get:
dni/dt = υi v = υi (dλ/dt)
And we obtain expression for the increase of entropy
in the system:
diS/dt = - 1/T Σ μi υi (dλ/dt) = - 1/T Σ μi υi v
As the reaction rate is equal for all components i, we get:
diS/dt = - 1/T (v) Σ μi υi
we introduce:
W = - Σ μi υi = (dG/ dλ)T,P
which represents work done by the system after all
reactions run through
and for the rate of entropy production we get:
diS/dt = Wv/T
This can be expressed in words:
the rate of entropy production in open system under
constant temperature and pressure is given by product of
reaction rate and the work done by the system.
Now we introduce expression for the rate of entropy
production in unit volume:
Θ = 1/V (dS/dt)
and function Ψ:
Ψ = TΘ
The functionΨ is proportional to the rate of entropy in
unit volume and is called dissipative function
We can rewrite it in general expression:
Ψ = T Θ = Σ J i Xi > 0
where Ji ...... rate of flux of the process
Xi ..... driving force of the process
Ψ depends on the rate of flux and driving force of the
process, which are time-dependent parameters, therefore
Ψ is also a function of time:
Ψ(t) = Σ Ji(t) Xi(t)
In equilibrium X equals J, as it holds under equilibrium
that X = 0, J = 0. We can thus assume that close to
equilibrium there is a linear relationship between fluxes
and forces and flux is a function of force:
J = J(X)
and it holds:
J(X) = L(X)
This equation represents linear phenomenological
relationship between the parameters of generalized
fluxes and forces, and the coefficient L is called
linear phenomenological coefficient
If we have a linear system close to equlibrium, we can write
expression for the rate of entropy production in the system:
diS/dt = 1/T Σ Jj Xj > 0
In agreement with II. law of thermodynamics this change
must be positive. Although the overall sum must be
positive, inside system can proceed one ore more processes
for which we can write:
diS'/dt = 1/T Σ Jk Xk ≤ 0
i.e. there are processes during which the entropy of
the system decreases
These conclusions in such a simple form holds true
only for linear relationships close to equilibrium. Living
systems, however, are nonlinear systems far from
equilibrium, in which irreversible processes proceed.
In agreement with II. law of thermodynamics any
irreversible process is accompanied by
heat of dissipation. In open system it is possible that this
heat leaves the system and the total entropy of the system
stayes constant, or even decreases.
We can write for the rate of entropy production in nonlinear systems:
dS/dt = deS/dt + diSn/dt + diSd/dt
where
diSn/dt ..... part of entropy production bound in the
system
diSd/dt ...... part of entropy production crossing the
boundaries of the system
Analogicaly with the preceding situation we have
function Ψ:
Ψ = Ψn + Ψd
where
Ψn ..... function of bound dissipation
Ψd ..... function of outer dissipation
According to the principle of the least outer dissipation
of energy:
In the stationary state of any thermodynamics system, the
function of outer dissipation reaches the least possible
values.
Physiology has been using long time the term, which is
very close to the concept of the stationary state in the
thermodynamics of irreversible processes. It is
basal metabolism.
Basal metabolism measured as the rate of heat production
or breathing represents the lowest metabolism of an animal
in rest. It is thus characterized by minimal rate of heat
production (minimum of the function of outer dissipation
of energy), that corresponds the concept of stationary
state.
The stationary state of living systems differs from the
stationary state of sipmple physical-chemical systems. This
difference consists in the fact, that in the simple systems the
stationary state is given by the outer parameters and stayes
stable only under maintained outer conditions. On the
other hand, living systems are able to resist the changes of
the outer environment by means of regulation and control
of the inner processes. Thus to describe the stationary
state of living systems it is more appropriate to use the
term homeostasis, introduced by Cannon.
As homeostasis we describe the ability of living organisms
to maintain the stability of inner medium during
occurence of random changes in the outer environment.
Living organisms are, from the point of view of
thermodynamics, open systems far away from thermodynamic
equilibrium. They are controlled and regulated. Exact
thermodynamic theory of such systems has not been
created yet.