Mathematical derivation for 1st law of thermodynamics :

What I know :

  1. 1st law of thermodynamics is known as or derived as the conservation of energy . But how are the terms U , q and W derived from it ?

  2. Another way it is derived on internet as:

The internal energy of a system can be increased in two ways. (i) By supplying heat to the system, (ii) By doing work on the system.

Then just combine these values. But how do you exactly know that there will be no other terms related to this proof ?

  1. http://web2.physics.miami.edu/~mbrown/PHY206_1stlaw.pdf . I couldn’t understand this proof.

If there is any simple way to derive the first law of thermodynamics. Please do share your answer.

  • $\begingroup$ Apply dimensional analysis either Buckingham or Rayleigh - all terms relevant stay. $\endgroup$
    – Solar Mike
    Jun 18, 2021 at 16:05
  • $\begingroup$ I suggest reading the history of the development of the first law to gain a better understanding of its original scope and later evolvement, which might have something you are looking for. BTW, I don't quite understand what part confuses you - the law of conservation of energy or the sources that produce heat during the heat transfer process? en.wikipedia.org/wiki/…. $\endgroup$
    – r13
    Jun 18, 2021 at 17:18
  • $\begingroup$ What do you want to derive it from? It is quite fundamental and one might simply consider it a postulate (experimentally verified). $\endgroup$ Jun 19, 2021 at 20:08
  • $\begingroup$ I suggest leaving proofs to the mathematicians and taking engineering equations as empirical. $\endgroup$
    – Tiger Guy
    Jun 21, 2021 at 13:20
  • $\begingroup$ @TigerGuy Hi. I am afraid that if i put it on maths stack Exchange, they will perhaps consider it as a scientific concept and not mathematical $\endgroup$
    – S.M.T
    Jun 21, 2021 at 16:08

3 Answers 3


Mathematical formulation of the first law of thermodynamics: (Relationship between internal energy, work and heat). The internal energy of a system can be increased in two ways. (i) By supplying heat to the system, (ii) By doing work on the system. Suppose the initial internal energy of the system = U1 If it absorbs heat q, its internal energy will become = U1 + q If further work w is done on the system, the internal energy will further increase and become = U1 + q + w. Let us call this final internal energy as U2. Then U2 = U1 + q + w or U2 - U1 = q + w or ΔU = q + w This equation is the mathematical formulation of the first law of thermodynamics.

  • $\begingroup$ You have written the same way as I wrote in my Q 2nd point. This isn’t derivation. $\endgroup$
    – S.M.T
    Jun 18, 2021 at 12:58
  • $\begingroup$ Then What’s the way you need… You asked for a simplified derivation right? $\endgroup$
    – Abarna
    Jun 18, 2021 at 13:16
  • $\begingroup$ For example . We know the derivation for ideal gas law as P directly proportional to Sth , then V , n , and T. Then , we just combined them and add the proportionality const. Right ? Is this the correct way of deriving an equation ? No. $\endgroup$
    – S.M.T
    Jun 18, 2021 at 13:17
  • $\begingroup$ chemistrygod.com/ideal-gas-equation-derivation. This is how you derive it. $\endgroup$
    – S.M.T
    Jun 18, 2021 at 13:18
  • $\begingroup$ So , I want a derivation similar to the formal of the link I sent just now. Hope you got it. If not , then do let me know. $\endgroup$
    – S.M.T
    Jun 18, 2021 at 13:18

Below shows the deviation of Q (heat), and W (work) in thermodynamic terms.

Through the law of conservative of energy, the first law of thermodynamics in derivative form can be expressed as:

$du = TdS - PdV$ (1)

From the definition of entropy (ref 1):

$Q = T\Delta S$

$\delta Q = TdS$ (2)

From pressure-volume work (ref 2):

$dV = \delta W/P$

$\delta W = PdV$ (3)

Plug (2) and (3) into (1):

$du = TdS - PdV = \delta Q - \delta W$

References for further studies:

  1. https://en.wikipedia.org/wiki/Heat

  2. https://en.wikipedia.org/wiki/Work_(thermodynamics)

  3. https://en.wikipedia.org/wiki/Conservation_of_energy

  4. https://en.wikipedia.org/wiki/Heat_transfer_physics


The internal energy of a system can be increased in two ways. (i) By supplying heat to the system, (ii) By doing work on the system. Then just combine these values. But how do you exactly know that there will be no other terms related to this proof ?

Consider a simple block of the type we'd envision for kinematics problems in introductory physics. It has no internal structure that matters. We could give it energy by raising it or by accelerating it, and we could measure its energy from any arbitrary reference height or speed. But whatever we do, the energy is expressed as just one number: 5 J, or 0 J, or -130 J, say. That's it.

Real objects are more complex. They contain, e.g., 1025 particles (molecules), each of which has its own energy. We could never catalog each value in this energy distribution. Fortunately, the distribution is often of a consistent general shape, and we need only two numbers to characterize it: the average or center value, say, and the breadth, or amount of variation or dispersion, say. This is similar to the way we describe statistical distributions such as the Gaussian, gamma, or lognormal distributions, for instance: by a location (perhaps the mean or median) and a width (perhaps the standard deviation or 95% confidence interval). For real objects near thermodynamic equilibrium, the location concept broadly corresponds to the energy, and the dispersion concept broadly corresponds to the temperature.

Stated again: if you have not just one energy but many, for a well-understood simple distribution, then you can describe the system in terms of just the distribution location and breadth. In turn, there are two ways you can alter this distribution: by changing its position (i.e., the energies of all particles in concert) or by changing the breadth (i.e., the width or dispersion of particle energies). The former is to do work on the system, and the latter is to heat it. When we do work on a system, we elevate all particle energies together; when we heat a system, we broaden the distribution of particle energies. This is the distinction between work and heat. Does this answer your question?

(Consider, for example, two very cold flywheels rotating in opposite directions. We suddenly press them together, causing them to brake to a halt and heat up in the process. In our frame, the linear and angular momenta remain zero before and after. The total mass stays the same, as does the total energy. But we've converted two very narrow peaks of energy—the energy of every molecule could initially be classified precisely by simply "clockwise" or "counterclockwise"—to a relatively broad distribution that is now amenable to description only by the equilibrium temperature, as the distinct particle information is now too vast to track. Colloquially, the work we did to start the wheels spinning has been converted entirely to heat.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.