# 2.3: Thermodynamics First Law

This law refers to conservation of energy in a non accelerating system. Since all the systems can be calculated in a non accelerating systems, the conservation is applied to all systems. The statement describing the law is the following.

\[

Q_{12} - W_{12} = E_2 - E_1

\label{thermo:eq:firstL} \tag{2}

\]

The system energy is a state property. From the first law it directly implies that for process without heat transfer (adiabatic process) the following is true

\[

W_{12} = E_1 - E_2

\label{thermo:eq:adiabatic1} \tag{3}

\]

Interesting results of equation (3) is that the way the work is done and/or intermediate states are irrelevant to final results. There are several definitions/separations of the kind of works and they include kinetic energy, potential energy (gravity), chemical potential, and electrical energy, etc. The internal energy is the energy that depends on the other properties of the system. For example for pure/homogeneous and simple gases it depends on two properties like temperature and pressure. The internal energy is denoted in this book as \(E_U\) and it will be treated as a state property. The potential energy of the system is depended on the body force. A common body force is the gravity. For such body force, the potential energy is \(mgz\) where \(g\) is the gravity force (acceleration), \(m\) is the mass and the \(z\) is the vertical height from a datum. The kinetic energy is

\[

K.E. = \dfrac{m U^2}{ 2}

\label{thermo:eq:ke} \tag{4}

\]

Thus the energy equation can be written as

Total Energy Equation

\[

\label{thermo:eq:largeEnergy}

\dfrac{m{U_1}^2}{2} + m\,g\,z_1 + {E_U}_1 + Q =

\dfrac{m{U_2}^2}{2} + m\,g\,z_2 + {E_U}_2 + W \tag{5}

\]

For the unit mass of the system equation (5) is transformed into

Specific Energy Equation

\[

\label{thermo:eq:smallEnergy}

\dfrac{{U_1}^2}{2} + g\,z_1 + {E_u}_1 + q =

\dfrac{{U_2}^2}{2} + g\,z_2 + {E_u}_2 + w \tag{6}

\]

where \(q\) is the energy per unit mass and \(w\) is the work per unit mass. The "new'' internal energy, \(E_u\), is the internal energy per unit mass. Since the above equations are true between arbitrary points, choosing any point in time will make it correct. Thus differentiating the energy equation with respect to time yields the rate of change energy equation. The rate of change of the energy transfer is

\[

\dfrac{DQ}{Dt} = \dot{Q}

\label{thermo:eq:dotQ} \tag{7}

\]

In the same manner, the work change rate transferred through the boundaries of the system is

\[

\dfrac{DW}{Dt} = \dot{W}

\label{thermo:eq:dotW} \tag{8}

\]

Since the system is with a fixed mass, the rate energy equation is

\[

\dot{Q} - \dot{W} = \dfrac{D\,E_U} {Dt} +

m\, U\, \dfrac{DU} {Dt} + m \dfrac{D\,B_f\,z} {Dt}

\label{thermo:eq:energyRate} \tag{9}

\]

For the case were the body force, \(B_f\), is constant with time like in the case of gravity equation (9) reduced to

Time Dependent Energy Equation

\[

\label{thermo:eq:energyRateg}

\dot{Q} - \dot{W} = \dfrac{D\,E_U} {Dt} +

m\, U \dfrac{DU} {Dt} + m\,g \dfrac{D\,z} {Dt} \tag{10}

\]

The time derivative operator, \(D/Dt\) is used instead of the common notation because it referred to system property derivative.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.