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Wikipedia: first law of thermodynamics

Laws of thermodynamics
Zeroth Law
First Law
Second Law
Third Law
Combined Law

The first law of thermodynamics is an expression of the universal law of conservation of energy, and identifies heat transfer as a form of energy transfer. The most common enunciation of the first law of thermodynamics is:

“

The increase in the internal energy of a thermodynamic system is equal to the amount of heat energy added to the system minus the work done by the system on the surroundings.

”

History

Main article: mechanical equivalent of heat

James Prescott Joule first laid down the foundation of the first law of thermodynamics, saying that heat and work are mutually convertible, through his extraordinary series of experiments.

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."

Mathematical formulation

The mathematical statement of the first law of a closed system is given by:

$\mathrm{d}U=\delta Q-\delta W\,$

where $d U$ is the infinitesimal increase in the internal energy of the system, $δ Q$ is the infinitesimal amount of heat added to the system, and $δ W$ is the infinitesimal amount of work done by the system on the surroundings. The infinitesimal heat and work are denoted by δ rather than d because, in mathematical terms, they are inexact differentials rather than exact differentials. In other words, there is no function Q or W that can be differentiated to yield δQ or δW.

The integral of an inexact differential is path dependent, i.e. it 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, (i.e. the integral is taken around a closed loop in thermodynamic parameter space) the value of the integral represents the change in the internal energy of the system.

Reversible processes

An expression of the first law can be written in terms of exact differentials by realizing that the work that a system does is equal to its pressure times the infinitesimal change in its volume. In other words, $δ W = p d V$ where $p$ is pressure and $V$ is volume. For a reversible process, the total amount of heat added to a closed system can be expressed as $δ Q = T d S$ where $T$ is temperature and $S$ is entropy. For a reversible process, the first law may now be restated:

$\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V\,$

In the case where the system is not closed, energy may also be brought into the system by the addition of new material. In this case the first law is written:

$\mathrm{d}U = \delta Q - \delta W + \sum_i \mu_i \mathrm{d}N_i\,$

where $d N i$ is the (small) number of type-i particles added to the system, and $μ i$ is the chemical potential of type-i particles.

$\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V + \sum_i \mu_i \mathrm{d}N_i\,$

Force-functions

A useful idea, introduced by Willard Gibbs in 1876, is that quantities such as internal energy U and Helmholtz free energy A may be regarded as a kind of force-function. For example, 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 and particle terms: $d U = p d V$ . The pressure p can be viewed as a force (and in fact has units of force per unit area) while $d V$ 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 is the amount of work-energy transferred as a result of the process.

It is useful to view the $T d S$ 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 is the amount of heat-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.

Sign convention

Physics and Chemistry

In physics and chemistry, the system is the object of greatest interest, and it is natural to talk about the work done on the system by the surroundings. This changes the sign of the equation. Defined in this manner, the first law is a generalization of this concept which states for a thermodynamic cycle that the net heat input is equal to the net work output. For a system with a fixed number of particles (closed system), the first law is stated as:

$\mathrm{d}U=\delta Q+\delta W\,$ ,

where

d U

is an infinitesimal increase in the internal energy of the system,

δ Q

is an infinitesimal amount of heat added to the system,

δ W

is an infinitesimal amount of work done on the system, and

δ

denotes an inexact differential.

Thermodynamics and Engineering

In thermodynamics and engineering, it is natural to think of the system as a heat engine which does work on the surroundings, and to state that the total energy added by heating is equal to the sum of the increase in internal energy plus the work done by the system. Hence $δ W$ is the amount of energy lost by the system due to work done by the system on its surroundings. During the portion of the thermodynamic cycle where the engine is doing work, $δ W$ is positive, but there will always be a portion of the cycle where $δ W$ is negative, e.g., when the working gas is being compressed. When $δ W$ represents the work done by the system, the first law is written:

$\mathrm{d}U=\delta Q-\delta W\,$

Very occasionally, the sign on the heat may be inverted, so that $δ Q$ is the flow of heat out of the system, and $δ W$ is the work into the system:

$\mathrm{d}U=-\delta Q+\delta W\,$

Because of this ambiguity, it is vitally important in any discussion involving the first law to explicitly establish the sign convention in use.

References

Goldstein, Martin, and Inge F., 1993. The Refrigerator and the Universe. Harvard Univ. Press. A gentle introduction.

External links

30+ Variations of the 1st Law
Mechanical Theory of Heat – Nine Memoirs by Rudolf Clausius [1850-1865] on the 1st and 2nd Laws of Thermodynamics.

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first law of thermodynamics