# Building a Definition for Heat

We start by examining the first law and defining each of it’s terms. The first law is simply a statement of the principle of conservation of energy and is often stated thus;

[tex]\Delta U = Q + W[/tex]

Where [itex]\Delta U[/itex] is the change in internal energy, Q is the heat added to the system and W is the work done on the system. So what do all these terms actually mean?

**Internal Energy**

We define the internal energy as the energy associated with the microscopic energies of system, that is with the energy associated with the random motion of the molecules within a system. So for a general fluid, the internal energy of a system is the sum of the translational kinetic energies, the rotational kinetic energies, the vibrational kinetic energies and the potential energies of all the molecules in that system. The internal energy of a system is often erroneously referred to as the heat of a system and we shall see why this is incorrect later. One important point to note here is that the internal energy is a state variable, that is, the change in internal energy between any two states is independent of the path taken.

**Work**

Well, if you’re reading this I assume that you know the definition of work; in thermodynamics work is usually associated with a transfer of energy into or out of a system. An example of work specific to thermodynamics would be the application of a force to a piston, which would then compress the gas within the cylinder, thus doing work on the gas. Since work is being done on the gas the W term in our expression would be positive. If we assume that the walls of the cylinder are adiabatic (no heat transfer) then all the work done would be converted to internal energy. Suppose that after we have compressed the piston, we release it. Intuitively, we would expect the piston to recoil back, and this is exactly what happens; the gas expands and does [an equal amount of] work on the piston against atmospheric pressure. In this case, since it is the gas that is doing work, our W term would be negative.

**Heat**

So we have defined the internal energy of a system and we can quantify the work done on the system, but what about heat? First let us examine temperature. One useful definition of temperature is often called kinetic temperature and is derived from kinetic theory. Using kinetic theory the temperature of a system is taken to be a measure of the average translational kinetic energy associated with the random motion of the molecules with the system. It should be noted that although related to internal energy, temperature is not directly proportional to internal energy since internal energy also involves the rotational and vibrational kinetic energies and the potential energies of the constituent molecules.

So, we come to the definition of heat. If we examine the first law, we can see that we can increase the internal energy of a system either by doing work on it, or adding heat to it. Consider a piston and a cylinder filled with gas, we can increase the internal energy of the system by either compressing the gas by applying a force to the piston (work) or by fixing the piston and placing the cylinder in a flame (heat). We can compress and heat the gas in such a way that after the operation all the macroscopic properties (pressure, volume & temperature) are identical, that is the two cylinders are in identical states. Suppose we take two identical cylinders (but not necessarily in identical initial states) filled with a gas at 373K, one of which we compress and the other of which we heat such that both cylinders are at 473K and all their macroscopic quantities are identical, that is the final states of the two cylinders are identical. If we were to now examine the final states of the two cylinders, we have no way of knowing which was compressed and which was heated; the only conclusion we can draw is that their internal energies have increased. In this way we can consider heat as the microscopic analogy of work (macroscopic).

I therefore, offer you a formal definition of heat:

“Heat is the non-mechanical exchange of energy between the system and surroundings as a result of a difference in temperature”

Both work and heat can be considered as methods of transferring energy within or between systems. It should now be apparent why the statement “a body posses heat” is nonsensical. To say that a body posses heat is analogous to stating that a “body has work”, which you must agree is utter rubbish. Rather, one transfers energy to a body by doing work on that body and one transfers energy to a body by heating or adding heat to that body. Similarly, it is incorrect to state that a body’s heat has increased, rather it’s internal energy has increased.

**A note about Thermal Energy**

Some texts make use of the term “thermal energy” when discussing the “translational kinetic energy” of the molecules, I personally find that the term “thermal energy” only serves to confuse discussions further.

Further Reading

Thermal Physics, 2[sup]nd[/sup] Edition, C.B.P. Finn

Heat@Hyperphysics

Internal Energy@Hyperphysics

Written by Hootenanny. Edited by berkeman and Kurdt.

We may not say "The body has work.", but it isn't unreasonable to ask how much work we can get out of a body. Saying a body has heat may be clear. Or it may need some extra precision to make the meaning clear. Which depends on the context and the audience. We create formal definitions so we can be precise when needed. Thank you for this definition.

Analysis on boundary, surrounding and system might clear out confusions. Both work and heat are boundary phenomena. There is no work if mass or energy does not cross over higher or lower system boundary.Potential energy is not work, but change in potential energy is Work. Like wise 500ton metals at 500 deg. C does not have heat energy unless there exist a difference of temperature in system(500 ton metal) and surrounding.

Caratheodory, who tried first to set thermodynamics on an axiomatic basis, tried to avoid the word "heat" altogether, but it is clear that the change of internal energy can be defined as the work done in an adiabatic process, i.e. ##\Delta U=W_\mathrm{adiabatic}##. Hence, heat is simply ##Q=\Delta U-W## where W is the work in a general non-adiabatic process. Adiabaticity can be defined before defining temperature by saying that the system is adiabatically isolated, if its state is independent of the surrounding. The real content of the first law is then that the adiabatic work starting from a reference state is a state function. This is a clear macroscopic definition of both internal energy and heat and one does not have to refer to microscopic degrees of freedom .

I therefore, offer you a formal definition of heat:“Heat is the non-mechanical exchange of energy between the system and surroundings as a result of a difference in temperature”How would the Peltier effect fit in this definition?

Thats not the point I wanted to make. You don't need external coils or the like for the Peltier effect. A heat flow may also be driven by purely mechanical forces without a temperature gradient. We know for more than 100 years by now (e.g. from the works of Pierre Curie in 1886) some basic principles of linear irreversible thermodynamics: General currents like heat current, mass current, electrical current, chemical reactions are driven by generalized forces like temperature gradients, pressure gradients, electrical potential gradients and chemical potential gradients. The important point is now that the linear relation between currents and forces is non-diagonal, i.e. in general, e.g. a pressure difference will not only drive a mass current but also a heat current ## \bf \rm even without the slightest temperature gradient. ##

“Suppose that after we have compressed the piston, we release it. Intuitively, we would expect the piston to recoil back, and this is exactly what happens; the gas expands and does [

an equal amount of] work on the piston against atmospheric pressure. ”This is not quite correct. It is only correct if both the compression and expansion are done reversibly, which certainly is not the case if expansion occurs adiabatically against constant atmospheric pressure.

In a closed system, no mass crosses the boundary of the system, but still, work can be done.

You are saying that heat cannot be transferred to a system unless there is a temperature gradient at the boundary, correct? Certainly, at the boundary, the temperature of the system must match the temperature of the surroundings (i.e., temperature is continuous at the boundary).

Yes, one example is sterling engine. Note that what I said was ” mass or energy”. Also my apology for stating “higher or lower system boundary.” It should be higher or lower system states.

Work can be done on a close system, given that boundary either expands or collapses, otherwise it’s useless. It’s like heating an LPG tank, no matter how much heat you apply on it, you can’t expect any work until it explodes.

No, it’s appropriate to say at the boundary the temperature is in between hot and cold reservoir (whichever is hotter – system or surrounding or vice versa)

Temperature is a continuous function of spatial position during an irreversible change, including at the interface between conductive solids and at the interface between real world reservoirs. However, the temperature gradient (heat flux) at the interface does not have to be continuous. Do you agree with this statement?

Chet

My apology Chet, your q is quite deep. I am not sure I got 100% of what you mean. Could you rephrase or give example, perhaps?

At time zero, you place a hot conductive semi-infinite solid slab in contact with an identical cold conductive semi-infinite solid slab, and let nature take its course. What do the temperature profiles look like within the two solids at times t > 0? Is the temperature a continuous function of spatial position, including at the boundary? Is the temperature gradient continuous at the boundary? Is the heat flux continuous at the boundary? What are the temperatures in the slabs far from the boundary? What are the the temperatures at the boundary?

Chet

The profile looks like this at the boundaries or the interface of two conducting surfaces. I’m actually lost by the term “continuous”, my apology “english” is not my native tounge.

Yes. This is more like it. I just wanted to clarify what you were saying. By continuous, what I mean is that the temperature does not change by a finite amount when one crosses the boundary between the two materials.

I tend to agree with the Starter, but before I read the other comments i would like to submit my composite view of work and heat by saying that both relate to transfer of energy between two systems. Heat is the transferred energy as a result of temperature difference and work is the transfer of energy where temperature is not directly involved. This kind of thinking for work and heat is most suited in the perview of Thermodynamics, which asserts that Heat and work areprocess variables and can be talked about only when two systems are involved. It release the historic mechanistic connection of work with force.

As DrDU talks of non-mechanical exchange of energy as heat, one should also consider non-mechanical work when electrical energy does the work on the system. I think that both heat and work should be dealt equivocally in the perview of Thermodynamics and then point out the essential difference. When an electric heater heats up we may losely say that electrical energy is converted to heat energy. But in the perview of Thermodynamics, Electrical energy is transferred to the heater coil increasing its internal energy raising its temperature above the surrounding which flows as heat to the surrounding. Similar arguments can be given for describing Peltier effect.

I beg to differ DrDu. “pressure difference will not only drive a mass current but also a heat current” here. does this statement not make heat the state variable? Heat Current means heat was here and now it is there!