In considering these two factors we may note that one factor is always a quantity (Q) and the other an intensity (I). This latter expresses some sort of difference of position or condition, the height of the weight, a difference of temperature in the heat engine, of pressure in the gas engine, or of electric potential in the dynamo or electric furnace. There can be no current of energy without this difference of potential, and therefore no transformation from one form of energy to another.
The second law of thermodynamics, Carnot's law, may therefore be enunciated thus: "Energy cannot be transformed without a fall of potential."
We may also derive this principle from a consideration of the formula of efficiency, the ratio of the work done by the transformer to the work done on the transformer.
Efficiency = energy transformed / total energy absorbed
The total energy is the product QI, i.e. the product of the total quantity by the total intensity at our disposal. The transformed energy is Q(I - I′), the product of the total quantity by the difference of intensity at the inlet and at the outlet of the machine. The formula for efficiency thus becomes
Q(I - I′) / QI = (I - I′) / I.
If I represents a temperature, then in order that the efficiency may be positive I′ must be less than I,
there must be a fall of temperature in the machine. If I′ were greater than I, i.e. if the temperature at the outlet were greater than that at the inlet, the efficiency would be a negative one, and the transformer would have to borrow heat from some external source.
Entropy.—In every transformation of energy a certain portion of the energy is transformed into heat: a lamp gives out useless heat as well as light, a machine gives out useless heat as well as mechanical work. This loss of useful energy as heat occurs in every transference or transformation of energy; it is only in the case of heat passing from a hotter to a colder body that there is no such transformation. When equality of temperature is established there has been no loss of energy, but the whole of the energy has become unutilizable, i.e. untransformable. In the formula of efficiency the fall of intensity I - I′ is now zero, and therefore the efficiency of the machine
(I - I′) / I