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The first law of thermodynamics

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The first law of thermodynamics

The first law of thermodynamics

What is the first law of thermodynamics, how is it stated mathematically, what does it mean and what are its implications? We discuss these and more.

We have previously seen the zeroth law of thermodynamics and established it as a common sensical, yet not readily obvious fact. We now discuss the first formally stated law of thermodynamics which comes courtesy of the German physicist, Rudolf Clausius. It was stated in 1850 in a completely different context from where it is applied now. (It was employed in the context of cyclic processes, now part of the more expansive second law.)

Conservation of energy

Conservation laws are the staple laws of physics; the state that the quantity of a given property in a predefined system (and by extension in the universe itself) is conserved, and neither newly created or unquestionably destroyed; it is merely interconverted between its alternate forms.

Energy can neither be created nor destroyed.

The law of conservation of energy states as much. In essence, energy in a system is constant but one form of energy (say, light) can become another (say sound or heat) but if you were to sum up the total energy in the system (as light, heat, whatever else) the sum is always constant at any given point of time. No energy ever goes missing or no extra energy magically appears.

The first law of thermodynamics

While there is a verbal statement of the first law of thermodynamics, it is not as implicative as the mathematical one and is almost never stated. It goes thus: “In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced.”

Which is just a fancy way of saying that the heat in a body is manifested as some energy of the body and some work done. Or, their algebraic sum is zero. Mathematically, the law becomes immensely more clear:

    \begin{align*} Q = \Delta U + W \end{align*}

where Q is the heat supplied, U is the internal energy of the system, and W is the work done by the system. Note the words in bold, because they are important factors in determining the signs we use. By simple convention, heat given out by the system will be -Q, whereas any work done on the system will be -W.

Work–energy interconversion

The units of measurement are apparent from the above formula: internal energy is measured in joule, as is the supplied heat and the work done. That would imply an equivalence of sorts between work and heat and energy. Indeed the examination of this equivalence is what led to the first law of thermodynamics.

Sir Humphrey Davy tried in vain to show this by rubbing ice blocks together, but it was fellow english physicist, James Prescott Joule, who would show the actual equivalence many years later, employing brass paddle wheels which were turned by falling weights, and then mercury replaced water and iron paddles replaced brass and Joule quickly realised that the change in state of the system was the same (depending only on the work done, therefore) regardless of the material.

It abolished the idea that heat and work and energy were some inherent properties, and established that work spends energy and gives rise to heat, which clearly would make a good, albeit crude, statement of the first law of thermodynamics.

The differential form

If the thermodynamic co-ordinates of a system change by infinitesimal quantities, in what is then called an infinitesimal process, the statement of the first law may be written more accurately as,

    \begin{align*} d Q = d U + d W \end{align*}

For processes so slow that they appear to be static (or occur infinitely slowly, in formal terms, which is an ideal case) — known as quasi-static processes — the terms dU and dW may be expressed in terms of other corresponding variables such as PVLT and so on, depending on the system.

For hydrostatic systems, for instance, we have d Q = dU + PdV, which makes our first law function in terms of PV and T. For more complex systems, additional work terms may similarly be added or subtracted (based on the convention discussed perviously).

In our next article we shall explore the concept of heat capacity, which leads on from the first law of thermodynamics, as well as talk about adiabatic and other types of processes categorised based on thermal exchange.

Cover image by Michel Filion.

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V.H. Belvadi is an Assistant Professor of Physics. He teaches postgraduate courses in advanced classical mechanics, astrophysics and general relativity. When he is free he makes photographs and short films, writes on his personal website, makes music, reads voraciously, or plays his violin. He currently serves as the Editor-in-Chief of Physics Capsule.

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