## Tuesday, February 15, 2011

### First law of thermodynamics

Thermodynamics

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The first law of thermodynamics is an expression of the principle of conservation of energy.
The law expresses that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed. It is usually formulated by stating that the change in the internal energy of a system is equal to the amount of heat supplied to the system, minus the amount of work performed by the system on its surroundings.

## Classical statement

The first explicit statement of the first law of thermodynamics was given by Rudolf Clausius in 1850:
"There is a state function E, called ‘energy’, whose differential equals the work exchanged with the surroundings during an adiabatic process."

## Description

The first law of thermodynamics says that energy is conserved in any process involving a thermodynamic system and its surroundings. Frequently it is convenient to focus on changes in the assumed internal energy (U) and to regard them as due to a combination of heat (Q) added to the system and work done by the system (W). Taking dU as an infinitesimal (differential) change in internal energy, one writes
$dU = \delta Q - \delta W\,$
where δQ and δW are infinitesimal amounts of heat supplied to the system and work done by the system, respectively. Note that the minus sign in front of δW indicates that a positive amount of work done by the system leads to energy being lost from the system.[note 1]
When a system expands in a quasistatic process, the work done on the environment is the product of pressure (P) and volume (V) change, i.e. PdV, whereas the work done on the system is -PdV. The change in internal energy of the system is:
$dU=\delta Q - PdV.\,$
Work and heat are due to processes which add or subtract energy, while U is a particular form of energy associated with the system. Thus the term heat for δQ means that amount of energy added as the result of heating, rather than referring to a particular form of energy. Likewise, work energy for δw means "that amount of energy lost as the result of work". Internal energy is a property of the system whereas work done and heat supplied are not. A significant result of this distinction is that a given internal energy change (dU) can be achieved by, in principle, many combinations of heat and work.
Informally, the law was first formulated by Germain Hess via Hess's Law, and later by Julius Robert von Mayer[1]

The classical statement of the first law of thermodynamics is induced from empirical evidence. It can be observed that given a system in an initial state, if work is exerted on the system in an adiabatic (i.e. thermally insulated) way, the final state is the same for a given amount of work, irrespective of how this work is performed.
For instance, in Joule's experiment, the initial system is a tank of water with a paddle wheel inside. If we isolate thermally the tank and move the paddle wheel with a pulley and a weight we can relate the increase in temperature with the height descended by the mass. Now the system is returned to its initial state, isolated again, and the same amount of work is done on the tank using different devices (an electric motor, a chemical battery, a spring,...). In every case, the amount of work can be measured independently. The evidence shows that the final state of the water (in particular, its temperature) is the same in every case. It's irrelevant if the work is electrical, mechanical, chemical,... or if done suddenly or slowly, as long as it is performed in an adiabatic way.
This evidence leads to the classical statement of the first law of thermodynamics
For all adiabatic processes between two specified states of a closed system, the net work done is the same regardless of the nature of the closed system and the details of the process.
This affirmation of path independence allows to define a state function, named internal energy, U, as the adiabatic work necessary to go from a reference state to a given one
$U(A)=U(O) + \tilde{W}^\mathrm{ad}_{O\to A}$
where, following IUPAC convention we take as positive the work done on the system.
To go from a state A to a state B we can take a path that goes through the reference state, since the adiabatic work is independent of the path
$\tilde{W}^\mathrm{ad}_{A\to B}=\tilde{W}^\mathrm{ad}_{A\to O}+\tilde{W}^\mathrm{ad}_{O\to B} = -\tilde{W}^\mathrm{ad}_{O\to A}+\tilde{W}^\mathrm{ad}_{O\to B} = -U(A) + U(B) = \Delta U$
In particular, if no work is exerted on a thermally isolated system we have
$\Delta U = 0\,$
or, in words,
The internal energy of an isolated system remains constant.
This is the law of conservation of energy.

### Diabatic processes

When the system does not evolve adiabatically, it is observed that the work exerted on the system does not coincide with the increase in its internal energy, which, being a state function, can be used for both adiabatic and non-adiabatic processes.
$\tilde{W}\neq \Delta U$
The difference is interpreted in terms of the heat that enters the system, so the inequality can be transformed in an equality as
$\tilde{W} + Q = \Delta U\,$
This is the usual expression the first law of thermodynamics. The inclusion of an unknown term (heat) does not transform it in a tautology, since its real physical content lies in the fact that there exists a state function that can be calculated independently of heat and work.

## State functional formulation

The infinitesimal heat and work in the equations above are denoted by δ, rather than exact differentials denoted by d, because they do not describe the state of any system. The integral of an inexact differential depends upon the particular path taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential will always be zero. The path taken by a thermodynamic system through a chemical or physical change is known as a thermodynamic process.
An expression of the first law can be written in terms of exact differentials by realizing that the work that a system does is, in case of a reversible process, equal to its pressure times the infinitesimal change in its volume. In other words δw = PdV where P is pressure and V is volume. Also, for a reversible process, the total amount of heat added to a system can be expressed as δQ = TdS where T is temperature and S is entropy. Therefore, for a reversible process:
$dU=TdS-PdV.\,$
Since U, S and V are thermodynamic functions of state, the above relation holds also for non-reversible changes. The above equation is known as the fundamental thermodynamic relation.
In the case where the number of particles in the system is not necessarily constant and may be of different types, the first law is written:
$dU=\delta Q-\delta W + \sum_i \mu_i dN_i\,$
where dNi is the (small) number of type-i particles added to the system, and μi is the amount of energy added to the system when one type-i particle is added, where the energy of that particle is such that the volume and entropy of the system remains unchanged. μi is known as the chemical potential of the type-i particles in the system. The statement of the first law, using exact differentials is now:
$dU=TdS-PdV + \sum_i \mu_i dN_i.\,$
If the system has more external variables than just the volume that can change, the fundamental thermodynamic relation generalizes to:
$dU = T dS - \sum_{i}X_{i}dx_{i} + \sum_{j}\mu_{j}dN_{j}.\,$
Here the Xi are the generalized forces corresponding to the external variables xi.
A useful idea from mechanics is that the energy gained by a particle is equal to the force applied to the particle multiplied by the displacement of the particle while that force is applied. Now consider the first law without the heating term: dU = − PdV. The pressure P  can be viewed as a force (and in fact has units of force per unit area) while dV  is the displacement (with units of distance times area). We may say, with respect to this work term, that a pressure difference forces a transfer of volume, and that the product of the two (work) is the amount of energy transferred as a result of the process.
It is useful to view the TdS  term in the same light: With respect to this heat term, a temperature difference forces a transfer of entropy, and the product of the two (heat) is the amount of energy transferred as a result of the process. Here, the temperature is known as a "generalized" force (rather than an actual mechanical force) and the entropy is a generalized displacement.
Similarly, a difference in chemical potential between groups of particles in the system forces a transfer of particles, and the corresponding product is the amount of energy transferred as a result of the process. For example, consider a system consisting of two phases: liquid water and water vapor. There is a generalized "force" of evaporation which drives water molecules out of the liquid. There is a generalized "force" of condensation which drives vapor molecules out of the vapor. Only when these two "forces" (or chemical potentials) are equal will there be equilibrium, and the net transfer will be zero.
The two thermodynamic parameters which form a generalized force-displacement pair are termed "conjugate variables". The two most familiar pairs are, of course, pressure-volume, and temperature-entropy.