# How can $U$ have an arbitrary zero point and still be extensive?

I'm reading Thermodynamics and an Introduction to Thermostatistics by H. B. Callen. Near the end of section 1-4 he writes:

Only differences of energy, rather than absolute values of the energy, have physical significance, either at the atomic level or in macroscopic systems. It is conventional therefore to adopt some particular state of a system as a fiducial state, the energy of which is arbitrarily taken as zero. (1)

He soon continues:

Like the volume and the mole numbers, the internal energy is an extensive parameter. (2)

Consider two systems with internal energies $$U_1$$ and $$U_2$$, respectively. According to (2) the internal energy of the joint system can be written $$U = U_1 + U_2$$. Statement (1) tells us that the fiducial states against which $$U$$, $$U_1$$ and $$U_2$$ are measured are arbitrary, so we should be able to change either one of them without affecting the physics. To keep it as general as possible, let us allow for changing all three and see what constraints we must impose stay true to (1) and (2).

$$U \to U' = U + x,$$ $$U_1 \to U'_1 = U_1 + y,$$ $$U_2 \to U'_2 = U_2 + z.$$

Now, (1) tells us differences of energies have physical significance, so these must be preserved by this transformation. Thus, on the one hand we have

$$U' - U'_1 = U - U_1,$$

but on the other hand we have

$$U' - U'_1 = U + x - U_1 - y,$$

so $$x = y$$. A similar argument with $$U'-U'_2$$ gives

$$x=y=z.$$

However, (2) still implies that $$U'=U'_1+U'_2$$, so

$$U_2 = U - U_1 = \left\{\text{differences preserved}\right\} = U' - U'_1 = U'_2 = U_2 + z$$ $$\implies x = y = z = 0.$$

We see that the only allowable transformation is no transformation at all! But then fiducial states are not arbitrary, contradicting (1).

Possible solutions

• Maybe Callen means something less general when he says that "Only differences of energy [...] have physical significance"? For example, he might mean differences of energy for the same system, or changes. I.e., quantities like $$\Delta U = U^f - U^i$$, but not quantities like $$U-U_1$$ in the above discussion.
• Maybe the correct transformation has nonlinear terms, unlike the family of transformations considered above. I have a hard time seeing how that could work, however.
• "Statement (1) tells us that the fiducial states against which $U$ $U_1$ and $U_2$ are measured are arbitrary, so we should be able to change either one of them without affecting the physics". Are you implying a different zero state can apply to $U_1$ and $U_2$? Sep 30, 2020 at 12:35
• @BobD I'm actually not sure, because I don't think Callen makes that entirely clear. Anyway, I think the argument above would work the same under either interpretation. It seems like all that would happen if $U_1$, $U_2$, and $U$ are measured against the same fiducial state is that we get additional conditions on $x$, $y$, $z$.
– ummg
Sep 30, 2020 at 16:57

You have taken the puzzle incorrectly backwards in a sense. The first sentence categorically says that the math that must apply to work with internal energy must apply exactly as below:

$$\Delta U_1 = U_B - U_A = \Delta U_2 = (U_B + U_o) - (U_A + U_o)$$

In this mathematical language, $$U_o$$ is one of two things. Either it is an energy that is added to both systems or it is an offset due to the measuring device.

Let's take the first case. Before we add any energy, combine the two systems as $$U_{To} = U_A + U_B$$. Now, add $$U_o$$ to this combined system to obtain $$U_{T1} = U_A + U_B + U_o$$. Now, let's split the new system. In order to remain true, we cannot MULTIPLY or ADD ANY OTHER energy during the process that we split the system. We must also split $$U_o$$ in the same way that we split back to $$U_A$$ and $$U_B$$. This results in

$$U_{T1} = U_{To} + U_o = (U_A + f_A U_o) + (U_B + f_B U_o)$$

In this, $$f_A$$ and $$f_B$$ are the proportions that go to A and B respectively. Because $$f_A + f_B = 1$$, we cannot make any combination of systems where $$f_A = 1, f_B = 1$$ simultaneously.

You might think about this with voltages (V) or electrical potentials (eV). You are trying to argue that adding 5 V to a 12 V source is the same as adding 5 V each to an 8 V and a 4 V source, then adding the two together. In other words, you are arguing that this should be true:

$$12 + 5 = (8 + 5) + (4 + 5)$$

You are stating that you can split one voltage source that is offset by 5 V into two voltage sources in parallel that sum to the original voltage and that are each individually offset by 5 V also with both offsets adding in parallel. You are doing so because you have heard that current flow does not depend on the absolute voltage of an object but only on the difference in voltage between two objects. So, why not just increase the voltage on all three states by a fixed amount (5 V)? As you might now see, this approach reverses the required math in exactly the wrong way.

By analogy, heat flow and work (flow) between two systems do not depend on the absolute internal energy of one system or another. Those energy flows depend only on the difference in internal energy between the two systems.

You can also make an analogy with ladders. You have two ladders that are 8 m and 4 m high respectively as measured from the first floor of the building. When you drop a ball from either, the results will give two different end point kinetic energies. The difference is a height of 4 m (in kinetic energy). Now, stack the ladders while you are on the first floor. You get 8 m + 4 m = 12 m. Now take the ladders to the second floor to drop a ball all the way back to the first floor level. Stand them separately. The difference is still 4 m (in kinetic energy). What you are trying to argue is that, when you take the ladders to the second floor of the building and stack them, you should be able to stack them in this way:

(8 m ladder + floor) + (4 m ladder + floor) = (12 m ladder + floor)

Of course, this is nonsense. We do NOT add the floor twice when we stack the ladders. But this is how you argue from a (mis)-interpretation of Callen.

You might instead be asking about the physical significance of a system truly at $$U$$ versus one truly at $$U + Uo$$ versus one truly at $$U$$ but measured by a device with a set offset of $$Uo$$ to give an apparent value of $$U + Uo$$. The latter case is what is meant by the second sentence in the first quote. Taking a particular state as having an arbitrary zero value is setting the zero point offset of a system according to our measuring tool. Although we change the offset of the measuring device that we use, we do not change the absolute energy content of the system. Setting a system to have a zero arbitrary energy is absolutely not the same as adding or removing energy to make the energy of that system zero. In conclusion, the seeming contradiction that $$U + Uo \neq U$$ measured with a device having an offset $$Uo$$ is solved by realizing that we cannot use two different measuring devices to compare the true state of any two systems.

Finally, one other way that the contradiction can be avoided is when we focus the first statement explicitly this way.

For any process undertaken on a single system, only differences of energy between the ending point and the starting point, rather than absolute values of the energy in either the starting or the ending point, have physical significance, either at the atomic level or in macroscopic systems.

Your step to combine two different systems violates the principles behind these unstated phrases. While this is an odd way out, it is still possibly a valid one depending on the larger context around the first statement.

In conclusion, the contradiction is to believe that shifting a zero point of a system is the same as adding/removing energy from a system. These are not the same. Callen is certifying that we can shift the zero point of our measuring frame because differences between systems are the only physically meaningful quantity. He absolutely is not certifying that we can arbitrarily add/remove a specific amount of energy to two different systems, combine them, and expect the summation is only shifted by that same amount of energy. As shown by the examples given here, the belief would lead to a gross, inverted violation of first principles based entirely on the mathematics alone. The counter that such a belief is categorically false is dutifully shown across a host of other physical phenomena where differences rather than absolutes drive the macroscopic or fundamental physical processes.

• Thank you for responding! I am not quite convinced by your first argument; essentially because I think there is a difference between adding $U_0$ to a system and changing the zero-point of the energy scale itself (which is what I am talking about). And of course, such a transformation cannot have the undetermined quantities $f_A$ and $f_B$.
– ummg
Sep 25, 2020 at 0:10
• Your voltage analogy is interesting in that it brings up the difference between voltage and electric potential. The zero-point of electric potential can of course be changed arbitrarily, but a voltage, being a difference of potentials, is unchanged by such a transformation. Maybe something similar is going on here? But I can't quite see which quantities are the voltages and which ones are the potentials. If $U=U_1+U_2$ is analogous to a voltage, while $U_1$ and $U_2$ are analogous to potentials, it places these quantities on unequal footing, which is problematic.
– ummg
Sep 25, 2020 at 0:18
• Now the final part of your answer, where you modify Callen's first statement, seems more promising (to me). It is similar to what an earlier commenter said (though that comment seems to have been removed). Essentially, when Callen writes "Only differences of energy" he might mean changes, rather than differences in general. Extensivity might then just reflect the fact that internal energy can be transferred from one system to another, unlike for example pressure. This would make my expression $U=U_1+U_2$ for the joint system nonsensical in the first place.
– ummg
Sep 25, 2020 at 0:30
• You still seem to need to justify that what you see as a contradiction must meant that the fundamentals contradict. In truth, your mathematics are an invalid translation of the opening statement. We are allowed (and indeed must have) a fractional amount of $U_o$ that goes to system A when we split the combined system or the math is invalid. Voltages are relative potentials (e.g. relative to ground). We can also change their zero points. Extensive properties depend on amount of substance. Scaler properties such as $T$ or $p$ or intensive properties (e.g. molar internal energy) do not. Sep 25, 2020 at 2:11
• I will have to ponder the first part of your response. When it comes to voltage on the other hand, you seem to have gotten it backwards. The voltage between point $A$ and point $B$ is defined as the work per charge in moving charge from point $A$ to point $B$. The electric potential at a point $P$ is the voltage between the reference potential (ground) and $P$. (Equivalently, voltages are potential differences.) So it is the potential that is relative to ground, not the other way around. Check wikipedia or any standard textbook if you doubt me.
– ummg
Sep 25, 2020 at 3:17

Reading carefully what Callen writes in the same chapter, clarifies the source of troubles. I assume that Callen's book can be considered a reputable source. It is true that

Only differences of energy, rather than absolute values of the energy, have physical significance, either at the atomic level or in macroscopic systems.

However, what is the energy he was speaking about? Let's confine the discussion to macroscopic systems. Macroscopic (internal) energy is defined by Callen, in the first chapter of his book, as the adiabatic work necessary to connect two equilibrium states, accordingly to a well established tradition in thermodynamics. Formally, one could write $$U(X_B) = U(X_A) + \int_{X_A}^{X_B}{\rm d}xW_{ad}(x)$$ i.e. the internal energy of the generic state $$B$$ is defined as the internal energy of the reference state $$A$$ (which is arbitrary) plus the adiabatic work along the transformation. It is the definition of adiabatic work which ensures that the integral does not depend on the path, but only on the initial ad final state.

But, let's look more carefully how each state is described. One independent variable should be the value of the variable the work $$W_{ad}$$ depends on. In this example it is $$x$$. In addition to this varible, a thermodynamic system is characterized by other variables, which are kept fixed in the adiabatic process. In particular, among the variables, some represent the size of the system. This is the case of the number of moles or the number of molecules ($$N$$).

Rewriting the previous formula with the explicit dependence on $$N$$, we have: $$U(X_B,N) = U(X_A,N) + \int_{X_A}^{X_B}{\rm d}xW_{ad}(x,N).$$ Therefore, it is true that only differences of energy have physical significance, because the difference between two generic states with the same number of molecules, $$U(X_B',N)-U(X_B,N)$$ does not depend on the value of the reference energy $$U(X_A,N)$$. However, the explicit argument $$N$$ shows that it is not possible to arrive to the same conclusion if the two states correspond to a different number of molecules: in that case, the difference $$U(X_B',N')-U(X_B,N)$$ depends explicitly on the difference $$U(X_A,N')-U(X_A,N)$$.

Indeed, this point was clear to Callen since, at the end of section 1-7 (I am referring to the second edition of his book) he stated explicitly:

...the methods of mechanics permit us to measure the energy difference of any two states with equal mole number.

Two paragraphs after this sentence, he added a recipe for relating the energies of states with a different number of moles: the energy of a compound system is defined as the sum of the original subsystems.

This implies that once one has arbitrarily chosen the zero of the energy of each isolated subsystem, it is not allowed to vary the zero of energy of the compound system. More formally, if energy of a subsystem $$1$$ is defined as $$U(X_{1B},N_1) = U(X_{1A},N_1) + \int_{X_{1A}}^{X_{1B}}{\rm d}xW_{ad}(x,N_1)$$ and similarly for sybsystem $$2$$, the energy of the compound system is $$U(X_{1B},N_1) +U(X_{2B},N_2)= U(X_{1A},N_1) + U(X_{2A},N_2) + \int_{X_{1A}}^{X_{1B}}{\rm d}xW_{ad}(x,N_1) +\int_{X_{2A}}^{X_{2B}}{\rm d}xW_{ad}(x,N_2)$$ and the energy of the reference state of the compound system $$U(X_{1A},N_1) + U(X_{2A},N_2)$$ cannot be changed at will. It has to stick to the (arbitrary) choice of the energy of the reference states of the subsystems. Said in another way, it is not allowed to vary independently the zero of energy of each subsystem and that of the compound system.

• Thank you for responding! To me, this answer seems most likely to be correct. It also supports my hypothesis that Callen's statement about "differences of energy" must be restricted to certain types of differences (here between states with the same $N$). I suppose you chose to single out $N$ because Callen does not provide a means for changing the composition of a system, while he does provide methods for traversing the $P$-$V$ plane?
– ummg
Oct 3, 2020 at 20:48
• About the restriction on $N$, Callen says: "Consider two simple subsystems separated by an impermeable wall and assume that the energy of each subsystem is known (relative to appropriate fiducial states [...]). If the impermeable wall is removed, the subsystems will intermix, but the total energy of the composite system will remain constant. Therefore the energy of the final mixed system is [...] the sum of the energies of the original subsystems. This technique enables us to relate the energies of states with different mole numbers.". Can this be formalized in the framework of your answer?
– ummg
Oct 3, 2020 at 21:26
• @ummg Yes, the starting point, when Callen introduces differences of energies , is the case of states with the same $N$. At some point he introduces the way to combine energies of subsystems. At that point the arbitrary choice of the zero of energy of each subsystem cannot imply the same arbitrariness for the compund system. I have added a final part to my answer to clarify this point. Oct 4, 2020 at 6:30
• Brilliant! Just one detail. For the fiducial states in addendum; should we not have $X_{1A}=X_{2A}\equiv X_A$, to keep it concistent with earlier parts of your answer? Otherwise the difference $U(X_{1B}, N) - (X_{2B}, N)$ (note the identical $N$) is ambiguous. Or is it your view that such a difference - between two different subsystems - is meaningless?
– ummg
Oct 4, 2020 at 17:28
• @ummg It does not matter: the second integral (from $X_{2A}$ to $X_{2B}$) can be rewritten as the integral from $X_{1A}$ to $X_{2B}$ minus the integral from $X_{1A}$ to $X_{2A}$. Tis last integral is $U(X_{2A})-U(X_{1A})$. Thus we end up with the same equation we would have by using from the beginning $X_{1A} = X_{2A}$. Oct 4, 2020 at 21:14

The problem is that shifting the zero-point of the energy does not always translate to the same shifts for each sub system as well as the entire combined system.

To make this explicit with a simple example, let's say you have two point masses of $$m_1$$ and $$m_2$$ in a uniform gravitational field. Then $$U_1=m_1gh_1$$ and $$U_2=m_2gh_2$$. Now let's change the zero-point. Then $$U_1'=m_1g(h_1-h_0)$$ and $$U_2'=m_2g(h_2-h_0)$$. Notice how the energy for mass 1 changed by $$m_1gh_0$$ and the energy for mass 2 changed by $$m_2gh_0$$, and the total energy changed by $$(m_1+m_2)gh_0$$. This is consistent with considering shifting the zero-point for the entire system.

Therefore, the transformation you describe is not correct in general, and this is where the issue lies. Changing the zero-point does not mean adding the same energy amount to all energies. It means changing where $$U=0$$, and so the corresponding energies then change accordingly by shifting appropriate amounts based on parameters of each system.

This statement you quote

Only differences of energy, rather than absolute values of the energy, have physical significance, either at the atomic level or in macroscopic systems.

refers to differences in energy across space which determine the forces. These differences are not affected by shifting the zero-point, since the shift effects all points in space equally.

• I think you are just restating the contradiction in slightly different terms. I stated it in terms of an inequality of energy differences because Callen emphasizes that only differences of energy carry physical significance. You instead stated it as $U'_1+U'_2\neq U'$. My point is that statement 1 implies that the zero point of the energy scale can be moved, transforming the internal energy of ANY system according to $U\to U'=U+U_0$. This contradicts statement 2 about extensivity. Thus is seems we must abandon extensivity, or else decide on a zero point for energy.
– ummg
Sep 23, 2020 at 22:12
• @ummg No, it's not a contradiction. You are assuming that shifting the scale for the entire combination is the same thing as shifting by the same amount for each system individually. This is not the case. Sep 23, 2020 at 22:13
• But $U$ is just a system, and we can refer to it without reference to (or even knowledge of) $U_1$ and $U_2$. So let us ignore, for the moment, $U_1$ and $U_2$. If we now change our energy scale to have its zero an amount $U_0$ lower than before we get $U' = U + U_0$ in the new, primed, scale. Otherwise, what is the correct way to transform $U$, without reference to $U_1$ and $U_2$? Are you saying that the correct way to move the zero point is some kind of nonlinear transformation? That seems like it would be difficult to make consistent with the physical significance of energy differences.
– ummg
Sep 23, 2020 at 22:28
• @ummg I have edited accordingly Sep 23, 2020 at 22:29
• I didn't downvote, but I really feel this is a misinterpretation. It's not wrong, because spatial energy gradients are one kind of energy difference we can measure, but it seems clear to me that Callen is talking about something far more general. What he says would be true even for a zero-dimensional system.
– d_b
Sep 24, 2020 at 3:24

Callen's statement is best interpreted as a statement about the units we choose to measure the energy of our system. He's saying that because only differences in energy are physically meaningful, the reference zero of our system of units can be set at an arbitrary point. But once we choose our reference zero, we have to stick to it; we can't change it willy-nilly in the middle of the problem.

I think the OP's confusion here is a matter of poor phrasing on Callen's part. When Callen says that the zero-point of our energy units is pegged to "a particular state of a system," it sounds an awful lot like he's saying every individual system gets its own zero point. But what he means is something more like:

• To calibrate his "Callen units", Callen selects a particular reference or "fiducial" state, "State $$0$$", of a particular thermodynamic reference system, "System $$0$$", and declares that its energy is zero.

• In these "Callen units", any other state, say State $$X$$, of System $$0$$ is assigned an energy value relative to that zero point. State $$X$$'s energy value, in "Callen units", is the change in energy that would be required to move from the reference state, State $$0$$, to the other state that we're considering, State $$X$$. (At this point, Callen also presumably scales his units by selecting a "fiducial" state of System $$0$$ which has an energy value of 1.0000... in "Callen units".)

• When Callen wants to measure the energy values of a state, say State $$A$$, of some other system, System $$Y$$, in his units, he takes System $$Y$$ in State $$A$$ and places it in contact with System $$0$$. He then tweaks the states of System $$0$$ till he finds one, State $$B$$, so that System $$Y$$ in State $$A$$ is in thermodynamic equilibrium with System $$0$$ in State $$B$$. The energy of State $$A$$ of System $$Y$$ in "Callen units" is then defined to equal the energy of State $$B$$ of System $$0$$.

Notice that Callen does not arbitrarily get to reset the zero point of "Callen units" when he starts considering a new system. System $$Y$$ will have "zero Callen energy" in State $$Z$$, exactly when System $$Y$$ in State $$Z$$ is in thermodynamic equilibrium with System $$0$$ of State $$0$$. So in your argument, it's true that Callen will have one fiducial state for $$U_1$$ and another one for $$U_2$$, but Callen doesn't get to pick these fiducial states independently: once he picks his fiducial state for $$U_1$$, the fiducial state for $$U_2$$ is automatically defined to be the state of $$U_2$$ which is in thermodynamic equilibrium with the fiducial state of $$U_1$$.

To keep it as general as possible, let us allow for changing all three and see what constraints we must impose stay true to (1) and (2).

$$U \to U' = U + x,$$ $$U_1 \to U'_1 = U_1 + y,$$ $$U_2 \to U'_2 = U_2 + z.$$

I don't think this is "as general as possible". Picking a zero point for my units on one of $$U_1$$, $$U_2$$ may have different effects than just adding a constant overall. Re-zeroing does make one of these shift by a constant, but that doesn't necessarily correspond to the other system's energy shifting by a constant, or the system as a whole.