The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states:

Description

Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. Heat is a process which transfers energy as a result of a temperature difference between a system and its surroundings. Mechanical work is the product of the force acting on a system and the distance moved in the direction of the force. However, work input can produce a temperature rise that is a mechanical equivalent of heat. 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 usually stated as:

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

where

$\backslash mathrm\{d\}U$ is an infintesimal increase in the internal energy of the system,

$\backslash delta\; Q$ is an infintesimal amount of heat added to the system,

$\backslash delta\; W$ is an infintesimal amount of work done on the system, and

$\backslash delta$ denotes an inexact differential.

As an analogy, if heat were money, then we could say that any change in our savings ($\backslash mathrm\{d\}U$) is equal to the money we put in ($\backslash delta\; Q$) minus the money we spend ($\backslash delta\; W$).

The δ's before the heat and work terms are used to indicate that they describe an increment of energy which is to be interpreted somewhat differently than the $\backslash mathrm\{d\}U$ increment of internal energy. In mathematical terms, heat and work are not exact differentials. Work and heat are processes which add or subtract energy, while the internal energy $U$ is a particular form of energy associated with the system. Thus the term "heat energy" for $\backslash delta\; Q$ means "that amount of energy added as the result of heating" rather than referring to a particular form of energy. Likewise, the term "work energy" for $\backslash delta\; W$ means "that amount of energy added as the result of work". The most significant result of this distinction is the fact that one can clearly state the amount of internal energy possessed by a thermodynamic system, but one cannot tell how much energy has flowed into or out of the system as a result of its being heated or cooled, nor as the result of work being performed on or by the system.

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."

Negative work: on or by the system?

In Chemistry and Physics, the system is the object of greatest interest, and it is natural to talk about the work done on the system. In 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 $\backslash delta\; W$ 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, $\backslash delta\; W$ is positive, but there will always be a portion of the cycle where $\backslash delta\; W$ is negative, e.g., when the working gas is being compressed. When $\backslash delta\; W$ represents the work done by the system, the first law is written:

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

Very occasionally, the sign on the heat may be inverted, so that $\backslash delta\; Q$ is the flow of heat out of the system:

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

Because of this ambiguity, it is vitally important in any discussion involving the first law to explicitly establish the sign convention in use. See The Absent-Minded Professor.

Mathematical formulation

The mathematical statement of the first law is given by:

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

where $\backslash mathrm\{d\}U$ is the infinitesimal increase in the internal energy of the system, $\backslash delta\; Q$ is the infinitesimal amount of heat added to the system, and $\backslash delta\; W$ is the infinitesimal amount of work done by the system. The infinitesimal heat and work are denoted by δ rather than d because, in mathematical terms, they are not exact differentials. In other words, 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 state space 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 equal to its pressure times the infinitesimal change in its volume. In other words $\backslash delta\; W=p\backslash mathrm\{d\}V$ where $p$ is pressure and $V$ is volume. For a reversible process, the total amount of heat added to a system can be expressed as $\backslash delta\; Q=T\backslash mathrm\{d\}S$ where $T$ is temperature and $S$ is entropy. For a reversible process, the first law may now be restated:

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

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:

$\backslash mathrm\{d\}U\; =\; \backslash delta\; Q\; -\; \backslash delta\; W\; +\; \backslash sum\_i\; \backslash mu\_i\; \backslash mathrm\{d\}N\_i\backslash ,$

where $\backslash mathrm\{d\}N\_i$ is the (small) number of type-i particles added to the system, and $\backslash mu\_i$ 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. $\backslash mu\_i$ is known as the chemical potential of the type-i particles in the system. The statement of the first law for reversible processes, using exact differentials is now:

$\backslash mathrm\{d\}U\; =\; T\backslash mathrm\{d\}S\; -\; p\backslash mathrm\{d\}V\; +\; \backslash sum\_i\; \backslash mu\_i\; \backslash mathrm\{d\}N\_i\backslash ,$

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: $\backslash mathrm\{d\}U=p\backslash mathrm\{d\}V$. The pressure p can be viewed as a force (and in fact has units of force per unit area) while $\backslash mathrm\{d\}V$ 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 (work) is the amount of energy transferred as a result of the process.

It is useful to view the $T\backslash mathrm\{d\}S$ 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 (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.

Application for open systems

In open systems, matter may flow in and out of the system boundaries. The first law of thermodynamics for open systems states: the increase in the internal energy of a system is equal to the amount of energy added to the system by matter flowing in and by heating, minus the amount lost by matter flowing out and in the form of work done by the system. The first law for open systems is given by:

$\backslash mathrm\{d\}U=\backslash mathrm\{d\}U\_\{in\}+\backslash delta\; Q-\backslash mathrm\{d\}U\_\{out\}-\backslash delta\; W\backslash ,$

where U_in is the average internal energy entering the system and U_out is the average internal energy leaving the system

The region of space enclosed by open system boundaries is usually called a control volume, and it may or may not correspond to physical walls. If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of matter into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of matter out as if it were driving a piston of fluid. There are then two types of work performed: flow work described above which is performed on the fluid (this is also often called PV work) and shaft work which may be performed on some mechanical device. These two types of work are expressed in the equation:

$\backslash delta\; W=\backslash mathrm\{d\}(P\_\{out\}V\_\{out\})-\backslash mathrm\{d\}(P\_\{in\}V\_\{in\})+\backslash delta\; W\_\{shaft\}\backslash ,$

Substitution into the equation above for the control volume cv yields:

$\backslash mathrm\{d\}U\_\{cv\}=\backslash mathrm\{d\}U\_\{in\}+\backslash mathrm\{d\}(P\_\{in\}V\_\{in\})\; -\; \backslash mathrm\{d\}U\_\{out\}-\backslash mathrm\{d\}(P\_\{out\}V\_\{out\})+\backslash delta\; Q-\backslash delta\; W\_\{shaft\}\backslash ,$

The definition of enthalpy, H, permits us to use this thermodynamic potential to account for both internal energy and PV work in fluids for open systems:

$\backslash mathrm\{d\}U\_\{cv\}=\backslash mathrm\{d\}H\_\{in\}-\backslash mathrm\{d\}H\_\{out\}+\backslash delta\; Q-\backslash delta\; W\_\{shaft\}\backslash ,$

During steady-state operation of a device (see turbine, pump, and engine), the expression above may be set equal to zero. This yields a useful expression for the power generation or requirement for these devices in the absence of chemical reactions:

$\backslash frac\{\backslash delta\; W\_\{shaft\}\}\{\backslash mathrm\{d\}t\}=\backslash frac\{\backslash mathrm\{d\}H\_\{in\}\}\{\backslash mathrm\{d\}t\}-\; \backslash frac\{\backslash mathrm\{d\}H\_\{out\}\}\{\backslash mathrm\{d\}t\}+\backslash frac\{\backslash delta\; Q\}\{\backslash mathrm\{d\}t\}\; \backslash ,$

This expression is described by the diagram above.

See also

• Conservation of energy
• Laws of thermodynamics
• Perpetual motion

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

Laws of thermodynamics | atmospheric thermodynamics

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