Explaining internal energy from a macroscopic perspective

This question stems from the accepted answer to this question

Classical thermodynamics deals with and relates macroscopic quantities, like pressure, temperature, mass, volume, etc. These are the quantities to conceive which we don't need to consider the molecular nature of the system. These are the quantities that can be defined even by disregarding the molecular nature of the system.

As highlighted by one of the answerers in the linked question, the internal energy is also a macroscopic quantity. In all the textbooks that I have referred, the internal energy is always defined as the sum of all the microscopic forms of the energy of the system. So, I always thought of it as a microscopic quantity, but it's not. Even though I was assisted in the linked question, that internal energy is defined by first law of thermodynamics, I don't get how?

How did people get to know there is a quantity called internal energy which is a state property, even though they did not know things were made of atoms and molecules?

In a nutshell,

How can we make sense of internal energy macroscopically, without associating it with the 'microscopic' word?

-This link discusses something of the sort, but I still can't understand what it is trying to say, maybe someone else does, and can explain me?

• "These are the quantities that can be defined even by disregarding the molecular nature of the system." That's not quite true. An equation of state relates these quantities at equilibrium and that equation depends on assumptions at the microscopic level. Consider the ideal gas equation. It assumes the internal energy of the gas is kinetic only. For a real gas, where intermolecular (Van der Waals forces) a different equation applies. Commented Jun 22, 2022 at 12:26
• @BobD When I say define, I mean we can define them macroscopically. Mass, volume, temperature, and pressure all can be conceived macroscopically. Sure we can also define pressure and temperature microscopically but when these concepts were introduced they were defined macroscopically. All these terms were used even when nobody knew about atoms and molecules. So in the same way, internal energy that we understand now as the sum of microscopic energies would originally have been introduced without considering any molecular nature of the system. Commented Jun 22, 2022 at 12:32
• All I'm saying is you can't define the relationship between the macroscopic properties without consideration of the microscopic properties, that 's all. Feel free to disregard the comment. Commented Jun 22, 2022 at 12:36

The Joule historic experience measuring carefully how mechanical work can raise the temperature of water is the basic idea of internal energy in my opinion.

We know from mechanics that the work of the net force is equal to the variation of the kinetic energy of a body. In the experiment, if there were no water, the kinetic energy of the paddle wheels would increase, proportional to the work of the dropping weight.

The drag force of the water decreases the net force. Of course the work of that (smaller) net force is equal to the increase of the kinetic energy of the paddle wheels. But now we can say that the work of the drag force causes the increase of the water temperature.

By analogy with the work energy theorem, that work is proportional to the variation of a quantity that we call internal energy.

What you need really is a proper starter-course in thermodynamics, with a good textbook (and reader can guess which one I recommend!)

The logic goes as follows.

1. Various experiments (especially those due to Joule) imply that internal energy is a function of state (I'll explain this below), up to whatever precision the experiment achieved.

2. This leads one to propose that it is a universal law that there is a quantity, called energy, which is a function of state, for any given system. Such a law does not logically follow from (1). Rather, it is an intuition motivated by (1). Having proposed the law, one then reasons from it and one can deduce a large number of other things. These are then compared with experiment. The whole framework is found to be consistent with its own logic and with experimental results, so one has a good scientific model, and that is what science is all about.

3. Another way of stating the law, logically equivalent to (2), is to say that energy is conserved.

Now I will briefly expand on (1) and (2).

The starting-point is to define energy by observing that for some physical processes we know how to compute the quantity called energy. Here are some examples:

• gravitational potential energy $$m g h$$.
• elastic energy in a Hooke's law spring: $$(1/2) k x^2$$.
• electrical energy $$\int I V dt$$
• work done against pressure by a change in volume: $$p\, \Delta V$$

We first note that these are mutually consistent because of the observed inter-convertability of forms of energy in mechanics, a fact we refer to as conservation of energy.

Next we come to some macroscopic system such as a given mass of a fluid. We note that it has many different thermodynamic states, characterized by properties such as volume, temperature, pressure, density, salinity, and others. We arrange to keep all these properties fixed except two, say temperature and volume. Then we arrange to move the fluid around its state space without supplying or taking away any heat. This is done by providing thermal isolation to prevent heat flow, and then by providing or extracting energy in the form of work (quantified by one of the above formulae). It is found in such experiments that to go between any two given states, the amount of work which has to be done is the same, no matter what method is used or what sequence of intermediate states is followed (provided, as I said, there is no heat flow and the other variables such as mass are kept constant).

That is the fundamental observation here.

It can only be observed at an accuracy limited by the precision of the experiments.

But now we conjecture that it is indeed fully accurate. This conjecture is called the first law of thermodynamics. It then follows logically that we can define a quantity called $$U$$ in the following way. When the system state changes from $$A$$ to $$B$$ the quantity called $$U$$ changes by $$U_B - U_A = \mbox{amount of work required to move the state from A to B without heat exchange}$$ The conjecture (called the first law of thermodynamics) guarantees that this definition will work.

We now have a quantity, associated with the symbol $$U$$, which can be associated with each state of a system. The above can only be used to calculate differences in $$U$$ but this is ok. All we need to do is assign $$U=0$$ to some given state, and then the value of $$U$$ for all the other states can be obtained. We would also like to be able to consider processes involving heat exchange. This is done by defining a new quantity called heat, such that the heat exchange in a given process is given by $$Q = \Delta U - W$$ where $$W$$ is the work done. This definition is not circular because we already defined all the quantities on the right hand side without the need to measure heat.

The quantity with the symbol $$U$$ now also earns a name. It is called internal energy.

So that is how internal energy is defined in thermodynamics without the need to mention anything about the microscopic structure of the entities involved.

How did we perform the 'magic' of getting a definition of internal energy without any idea of the microscopic structure or nature of energy? The answer is by the combination of theory and experiment. And that is, of course, how all of science is done. In this example the first step in the theoretical 'method' is simply to propose the laws of thermodynamics as axioms. This is similar to Newtonian mechanics where one may take Newton's laws as axioms. The next step is to learn how to reason from the axioms in useful ways.

I think that science education in the last fifty years has led some confusion about the logical basis of thermodynamics. It is widely said to require a microscopic model, to which it is an approximation, valid in the thermodynamic limit. But in fact thermodynamics does not require any microscopic model. That doesn't make it either right or wrong; it is a useful generalization from experience, like every other scientific framework. The relationship between thermodynamics and microscopic models is that they should be mutually consistent in domains where their assumptions and methods both apply. If both are valid then they each provide a constraint on the other.

• a system can be taken from state 1 to state 2 by adding energy via work, irrespective of what type of work it is, as you said. How does that make me conclude that there is a state property like $U$? Is it because since I added energy to the system, in state 2, that energy must be stored 'internal' to it. I can even argue that the system had some energy stored internally even in state 1, which was given to it while bringing it from a prior state 0 to 1? Commented Jun 24, 2022 at 14:23
• @HarshitRajput For this you need to invoke some mathematical methods and proofs involving partial derivatives and the concepts of state function and proper differential. The main fact is that if an integral between given endpoints is independent of the path then there exists a single-valued function of state whose change is given by the integral. But as I say you need to learn the maths along with the physics. A good textbook will present all of this. Commented Jun 24, 2022 at 17:26
• That is so interesting. Is it something like what @JanLalinsky wrote in his answer? Commented Jun 25, 2022 at 4:45
• @HarshitRajput Yes. My answer simply expands on the same point he is making. Commented Jun 25, 2022 at 8:46
• Can you recommend any good book, in general, for thermodynamics? I'm currently referring Fundamentals of Engineering thermodynamics by Moran and Shapiro Commented Jun 28, 2022 at 6:18

What came up in comments to this answer to the question quoted in the OP, is that one has to distinguish, on the one hand, phenomenological vs. microscopic models/theories and on the other hand, macroscopic vs. microscopic scales/quantities. The meaning of word microscopic and what it is opposite to depends on the context (alas, human language is ambiguous - e.g., Sisyphus is an example of hard and futile work, even though he performs zero work.)

A phenomenological model is a scientific model that describes the empirical relationship of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. In other words, a phenomenological model is not derived from first principles. A phenomenological model forgoes any attempt to explain why the variables interact the way they do, and simply attempts to describe the relationship, with the assumption that the relationship extends past the measured values.

When applied to physical phenomena and bodies, the macroscopic scale describes things as a person can directly perceive them, without the aid of magnifying devices. This is in contrast to observations (microscopy) or theories (microphysics, statistical physics) of objects of geometric lengths smaller than perhaps some hundreds of micrometers.

Internal energy is a quantity that exists both in (phenomenological) thermodynamics, and (microscopic) statistical physics. This is a quantity describing a system with a huge number of particles, $$N\sim 10^{23}$$, and directly measurable - this is why, in terms of scale, it is a macroscopic quantity.

Remarks:

• Use of word phenomenological is again ambiguous - when applied to phenomenological models it does not mean just any model that describes a phenomenon.
• In nanotechnology it is rather common to speak about mesoscopic scale - the scale/systems where we still deal with many particles, but must take account of their microscopic behavior (which usually means taking into account their quantum properties). See, e.g., the classical book by Joe Imry: Introduction to mesoscopic physics.
• A good example is the ideal gas law that came in now deleted comments: In statistical mechanics the ideal gas equation is derived from microscopic principles. However, the phenomenological gas laws were known and successfully used (e.g., to construct steam engines) before the development of statistical physics. Adding adjective ideal does require microscopic theory.
• Oops! My comment was meant for the OP. And I'm only pointing out one should be aware of the microscopic assumptions Commented Jun 22, 2022 at 12:25
• @RogerVadim Thanks for your response. I'm not able to understand how this relates to my question. What I'm trying to get an answer to is, before anyone knew about atoms and molecules, how were they able to get to reach to the concept of internal energy. That is, how classical thermodynamics would explain internal energy? Commented Jun 22, 2022 at 13:17
• @HarshitRajput as energy stored in an object, which can be changed by heating it or performing work on it. en.wikipedia.org/wiki/Internal_energy Commented Jun 22, 2022 at 13:30

The concept of internal energy is introduced in the first law of thermodynamics, based on Joule's work. Based on his experiments with heating and mixing water with a paddle wheel, he realized the effect of heat added to the system on its macroscopic state (in his experiments, the temperature) can be achieved also by equivalent amount of work done on the system, and this effect can be "undone", i.e. the body returns to the original state, when the body releases equivalent amount of heat into environment. One calorie is equivalent to 4.2 Joule of work, so we can measure heat in Joules.

In a cyclic physical process, where the system undergoes changes but eventually returns to the original state, and both work and heat are exchanged with the environment, net sum of heat and work added to the system during the whole cycle is zero:

$$\Delta Q + \Delta W = 0 ~~(whole~cycle).$$

If the changes are done quasistatically, so that the system remains in equilibrium thermodynamic state during the whole cycle, the physical process is associated with a closed path in the space of equilibrium states $$\gamma$$, and we can write $$\oint_\gamma dQ + dW = 0.~~~(*)$$

Since this equation holds for any closed curve $$\gamma$$ (assumption generalizing Joule's observations), it defines function $$U$$ of equilibrium state, via

$$dU = dQ + dW.$$ How does this define function $$U$$? We choose some special state $$\mathbf{X}_0$$ where we define internal energy to have some value $$U_0$$ (this can be zero). Then we can define internal energy of any other state $$\mathbf X$$ as follows. We find one possible orientedpath $$\delta$$ in the space of equilibrium states going from $$\mathbf X_0$$ to $$\mathbf X$$, and then define

$$U(\mathbf X) = \int_\delta dQ + dW.$$

Which one of the many possible paths is used does not matter, because (*) guarantees that value of the last integral is always the same.