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# First law of thermodynamics

The most fundamental idea in thermodynamics is the conservation of total energy, which is termed the "first law" of thermodynamics. The first law is based on our every day observation that for any change of thermodynamic properties, total energy, which includes internal, potential, kinetic, heat, and work, is conserved.

## Open thermodynamic system

We begin with the first law of thermodynamics applied to an open thermodynamic system. As illustrated in Fig. 1, an open system allows mass and energy to flow into or out of the system. We make the following assumptions and definitions:

1. Mass flows into or out of the system along one boundary of the system. The mass flow rate into the system is positive, whereas flow rates out of the system to the surroundings are negative.
2. Mass can carry internal energy into or out of the system. We neglect kinetic and potential energy carried by the mass. This is often a good assumption when the fluid is not moving near the speed of sound, the change in height over the system is not large, or the system temperature variations are not large.
3. The only types of work that are present are expansion/compression of the system and flow work. The boundaries of the system can expand or contract. Thus, work can be done by the system on the surroundings or vice versa. Work is positive when the surroundings do work on the system (i.e., the system contracts). The mass that enters or exits the system also does work—sometimes called flow work or pressure work.
4. We neglect potential energy and kinetic energy changes within the system.
5. Energy in the form of heat might enter or leave the system across the system boundaries. Heat transfer is positive when heat is exchanged from the surroundings to the system.

Before proceeding, we must define internal energy. Internal energy of a substance is the sum of the potential energy arising from chemical bonds of atoms and electrons and the sum of the kinetic energy of the atoms and molecules. The microscopic kinetic energy is sometimes called thermal energy, which is proportional to temperature.

With this definition, a total macroscopic energy balance in the system, at an instantaneous point in time, gives ....................(1)

where ....................(2)

and nU, the total internal energy, is equal to the total energy within the system by assumption four previously discussed. The property, U, is the molar internal energy (total energy/mole). Eq. 2 shows that when work or heat is added to the system, the molecular activity increases, causing the total internal energy to increase; that is, .

The term on the right side of Eq. 1 contains three terms: mass influx into the system that carries energy; heat transfer into the system; and compression work done by the surroundings on the system. Because we neglect potential and kinetic energy of the mass that flows into the system (assumption two), the energy associated with the mass influx into the system is simply Uṅ, where is the molar flow rate. Based on assumption one, there is only one molar flow rate into the system. The rate of heat flow from the surroundings across the system boundaries into the system is given by . Compression (or expansion) of the system boundaries causes work on the system denoted by Ẇ. Substitution of these terms into Eq. 1 gives ....................(3)

where the left side refers to energy within the system and the right side to energy that flows across the system boundaries into the system.

The two types of work considered are expansion/compression work and flow work (assumption three). From physics, work is performed whenever a force acts over a distance. Thus, the differential mechanical work that results from a differential displacement is given by . For expansion/compression work, the external force is equal to an external pressure supplied by the surroundings multiplied by the corresponding area along the boundary of the system.

In Cartesian coordinates, dW = - pext(Axdx + Aydy + Azdz) , where the external pressure is constant along the boundary of the system. Ax is the area normal to the x -coordinate that is being displaced, and so forth. The minus sign indicates that work is positive if the displacement is negative (i.e., an external force compresses the system). The expression for the differential work can be simplified further as , where nV is the total volume and V is the molar volume (volume/mole). The rate of work is . This equation applies to any arbitrarily shaped system.

For example, consider a rectangular box that expands differentially into the surroundings on three sides, as illustrated in Fig. 7.4. Here, Axyz, Ayxz, and Azxy, where the differential cross terms are neglected. The differential work is, therefore, dW = -pext(yzdx + xzdy + xydz) , provided the external pressure is the same on all faces of the box. The differential displacement volume is equal to d(nV) = d(xyz) = yzdx + xzdy + xydz, which gives the desired result. In this example, the differential work is negative because the system does work on the surroundings. The example also illustrates that even though the system expands into the surroundings, the work is always related to the external pressure. If the external pressure is zero, no work will be done by the system because the surroundings will offer no resistance.

Flow work is done by mass that enters or exits the system. A flowing fluid element does work on the fluid ahead of it, and the fluid behind it does work on that fluid element. Flow work, for example, turns the turbine shaft of a hydroelectric power plant. For one-dimensional (1D) inflow or outflow of fluid, the instantaneous rate of flow work is , where u is the velocity of the fluid; ρ is the molar density (i.e., inverse of V); and the molar flow rate is = ρAu. Flow work is positive when the fluid is entering the system; that is, the surroundings do work on the system.

For reversible displacements, the pressure in the system must equal the external pressure, p = pext, supplied by the surroundings, so the system and surroundings are always in equilibrium. With the assumption of reversibility, the total rate of work (expansion/compression work plus flow work) becomes . Eq. 3 is then written as ....................(4)

Eq. 4 can be simplified by defining the enthalpy, H = U + pV. The definition for enthalpy is defined strictly for mathematical convenience. For liquids and solids at low pressures, we often take H = U because the product pV is small compared to U (the molar volume of the condensed phases is also small). Combining the first and third terms on the right side of Eq. 4 gives ....................(5)

Eq. 5 can be written in thermodynamic shorthand as ....................(6)

## Closed and isolated systems

For closed systems, dn = 0 (the total moles in the system is constant), and Eq. 6 becomes

For isolated systems, dn = 0, dQ = 0, and d(nV) = 0; therefore, Eq. 6 reduces to ....................(7)

which shows that the total internal energy of an isolated system is constant (nU=constant).

## Nomenclature

 H = molar enthalpy of fluid, energy/mole, J/mole n = total moles of all components, moles p = pressure, force/area, Pa Q = net heat transferred, energy, J U = molar internal energy, energy/mole, J/mole V = vapor mole fraction, moles vapor/total moles, dimensionless or molar volume of fluid, volume/mole, m3/mole W = net work performed, energy, J