# On the work of Sadi Carnot

This PDF file contains an English translation of Sadi Carnot's Refelections on the motive power of fire. It also contains commentary by Lord Kelvin. And some chapters about the work and life of Carnot, made by the editor R. H. Thruston.

Under the section, The Work of Sadi Carnot, the editor talks about how Carnot discovered what's called the principle of equivalence between energy and heat. He also talks about the units Carnot employed to measure energy and heat. He writes:

In making his measures of heat-energy, he assumes as a unit a measure not now common, but one which may be easily and conveniently reduced to the now general system of measurement. He takes the amount of power required to exert an energy equal to that needed to raise one cubic meter of water through a height of one meter, as his unit; this is 1000 kilogrammeters, taken as his unit of motive power; while he says that this is the equivalent of 2.7 of his units of heat; which latter quantity would be destroyed in its production of this amount of power, or rather work. His unit of heat is thus seen to be 1000 /2.7, or 370 kilogram meters. This is almost identical with the figure obtained by Mayer, more than ten years later, and from presumably the same approximate physical data, the best then available, in the absence of a Regnault to determine the exact values. Mayer obtained 365, a number which the later work of Regnault enabled us to prove to be 15 per cent. too low, a conclusion verified experimentally by the labors of Joule and his successors. Carnot was thus a discoverer of the equivalence of the units of heat and work...

My questions are:

-Carnot defined a unit of energy as that energy that raises a meter cube of water one meters high. This is equivalent to $1\text {kg} m^2s^{-2}=\text {joule}$. But the editor says this is "1000 kilogrammeters". I'm puzzled by this assertion, since kilogrammeters is not a dimensionally correct unit of energy. Also where does this $1000$ come from?

-Why Carnot uses a unit of heat that's not equal to a unit of energy (different by a factor of $2.7$), although he knew that energy and heat are the same physical quantities. Why not choose the same units for energy and heat?

Edit: I just learned from user Mauro that kilogram-meter=joule. In page 100, Carnot defined a unit of heat as the amount of heat needed to raise a body of one kilogram by 1 degree. The specific heat of water is 4.186 joule/gm.celsius or 4.186 kilogram-meter/gm.celsius. $Q=cm\Delta t$. So a unit of heat according to his definition is= 4.186 kilogram-meter/gm.celsius *1000 gm*1 celsius= 4186 kilogram-meter or 4186 joule (1kilo-calorie) .

So his definition of energy is energy needed to lift one kilogram one meter high which is equal to is 1 joule or 1 kilogram-meter(as stated in the quotation). So the ratio between his unit of heat and energy should be 4186/1=4186; a factor of 4186 not 2.7; so where does 2.7 come from? And where does the factor of 1000 stated by the editor come from?

• @MauroALLEGRANZA no no, I did not mean that, but he defined a unit of energy as the energy needed to lift one kilogram one meter high. This is equal to 1 joule. I will remove the term SI, as it caused confusion to you. By working in SI, I meant he used kg and meter as his units for mass and length. – Omar Nagib Sep 24 '15 at 20:41

Alright after doing some research I figured out the answers to all my questions.

1)Carnot's definition of energy is as follows: it's the energy needed to lift a cubic meter of water one meters high. By definition a cubic meter of water is 1000 kg, and following from the definition of work according to Carnot is: $W=Fd=mgh$ , where $g=9.8 ms^{-2}$ and $h=1 m$ therefore a unit of work = $1000 \text{kg}*9.8ms^{-2}*1 m=9800$ $\text{joule}$.

2) $\text{kilogrammeter}$ (also more correctly known as $\text{kilogram-force meter [kgf*m]}$) is a unit of work. $1$ $\text{joule}=0.101971621$ $\text{kilogrammeter}$; Therefore $9800$ $\text{joule}=999.3 \approx1000$ $\text{kilogrammeter}$.

3)Carnot's definition of unit of heat is stated in page 100 of the linked PDF as follows:

If, then, we take for the unit of heat the quantity necessary to raise 1 kilogram of water 1 degree

Which is = $1$ $\text{Kcal}$.

4)In a posthumous notes transmitted to the french academy of science, Carnot writes:

According to some ideas that I have formed on the theory of heat, the production of a unit of motive power necessitates the destruction of 2.70 units of heats.

Also thanks for the help presented by user Mauro.

The "heat-energy" [page 8] is a motive power, i.e.

work = force $\times$ distance

and the kilogram-meter is a unit of work.

• Yes, but why the editor says this is equal to 1000 kilogram-meters, not 1 kilogram-meters?. In addition to that why Carnot defines a unit of heat different from a unit of energy? – Omar Nagib Sep 24 '15 at 18:57
• I disagree with your last quotation having anything to do with the factor of $1000$ and $2.7$. So regarding the unit of heat, Carnot in page 100 states: "we take for the unit of heat the quantity necessary to raise 1 kilogram of water 1 degree". We know that $Q=cm\Delta T$. And the specific heat of water is $4.186$ joule/gm.celsius. So according to his definition: unit of heat=4.186 joule/gm.celsius *1000 gm *1 celsius=4186 joule. 0.267 is the ratio between the specific heat of water and air. And body $A$ happened to have $0.267$ units of heat(as defined by carnot), but this is irrelevent to – Omar Nagib Sep 24 '15 at 19:58
• The coeff. Of unit of heat having 2.7 that of unit of energy. And also I still don't know where does the 1000 factor come from – Omar Nagib Sep 24 '15 at 19:59