Modern Engineering Thermodynamics

4.3 The First Law of Thermodynamics 101
AN EXAMPLE OF MATHEMATICAL SYMMETRY
Here is a story about Carl Friedrich Gauss (1777–1855). When he was a young child, his teacher wanted to occupy him for
a while, so he asked him to add up all the numbers from 1 to 100. That is, find X =1+2+3+… + 100. To the teacher’s
surprise, Gauss returned a few minutes later and said that the sum was 5050.
Apparently Gauss noticed that the sum is the same regardless of whether the terms are added forward (from first to last) or
backward (from last to first). In other words, X =1+2+3+… + 100 = 100 + 99 + 98 + … + 1. If we then add these two
ways together, we get
X = 1 + 2 + 3 + … + 100
X = 100 + 99 + 98 + … + 1
2X = 101 + 101 + … + 101
So 2X = 100 × 101 and X = (100 × 101)/2 = 5050. Gauss had found a mathematical symmetry, and it tremendously
simplified the problem. What is conserved here? It is the sum, X. It does not change no matter how you add the numbers.
Table 4.1 Relation of Conservation Laws to Mathematical Symmetry
Conservation Law
Mathematical Symmetry
Linear momentum
Angular momentum
Energy
Electric charge
The laws of physics are the same regardless of where we are in space. This positional symmetry implies
that linear momentum is conserved.
The laws of physics are the same if we rotate about an axis. This rotational symmetry implies that angular
momentum is conserved.
The laws of physics do not depend on what time it is. This temporal symmetry implies the conservation of
energy.
The interactions of charged particles with an electromagnetic field remain the same if we multiply the fields
by a complex number e iφ . This implies the conservation of charge.
In summary, Emmy Noether’s theorem shows us that (Table 4.1)
■
■
■
■
Symmetry under translation produces the conservation of linear momentum.
Symmetry under rotation produces the conservation of angular momentum.
Symmetry in time produces the conservation of energy.
Symmetry in magnetic fields produces the conservation of charge.
4.3 THE FIRST LAW OF THERMODYNAMICS
In this chapter, we focus our attention on the detailed structure of the first law of thermodynamics. To completely
understand this law, we need to study a variety of work and heat energy transport modes and to investigate the
basic elements of energy conversion efficiency. An effective general technique for solving thermodynamics problems
is presented and illustrated. This technique is used in Chapters 5 and 6 and the remainder of the book.
The simplest, most direct statement of the first law of thermodynamics is that energy is conserved. That is, energy
can be neither created nor destroyed. The condition of zero energy production was expressed mathematically in
Eq. (2.15):
E P = 0 (2.15)
By differentiating this with respect to time, we obtain an equation for the condition of a zero energy production
rate:
dE P
= _E p = 0 (2.16)
dt
Whereas Eqs. (2.15) and (2.16) are accurate and concise statements of the first law of thermodynamics, they are
relatively useless by themselves, because they do not contain terms that can be used to calculate other variables.
However, if these equations are substituted into the energy balance and energy rate balance equations, then the
following equations result. For the energy balance,
E G = E T + E P ðas required by the first lawÞ