## Saturday, November 17, 2012

### Thermodynamics using Earth-centric Units

While the trend in science has been to point out the obvious fact that the Earth is not the center of the solar system or the universe, over the last few decades there has been a trend in the engineering community of trying to teach thermodynamics with an Earth-centric focus. The new quantity that has appeared in this Earth-centric thermodynamics is called exergy.

Exergy is defined as the maximum amount of useful work that can be generated by bring a non-equilibrium system into equilibrium with the Earth's environment, i.e. bringing the system to the same temperature, pressure and chemical composition of the Earth's environment. The Earth has an enormous amount of stored exergy (fossil and nuclear fuels) as well as an enormous amount of exergy supplied to it in the form of light from the Sun. In some ways, the definition of exergy is completely arbitrary because the amount of useful work that can be obtained in bring a system into equilibrium with Earth environment is always less than the amount of useful work that could be obtained by bringing a non-equilibrium system into equilibrium with a ~0 K vacuum. Exergy is a really useful engineering quantity, even though it has no scientific meaning (since somebody living on a planet with a different temperature and/or a different atmospheric pressure/composition would measure a different value of exergy for the same exact enclosed system.)

But since exergy is such a useful definition for power plant engineers, this begs the question: are there other ways that we can modify the way we teach thermodynamics so as it make thermodynamics easier to learn and to apply? (Note that this is also one of the reasons why I'm in favor of having a currency that keeps the average price of purchasing useful work a constant. It makes it significantly easier to teach and to learn thermo-economics, i.e. that currency represents your capability to purchase useful work.)

I've been working on a new set of units for some standard thermodynamic quantities (such as temperature and pressure) and a new zero point for other standard thermodynamic quantities (such as voltage.) There's a few reasons for developing a new set of units. Here's a list of some of the reasons to change:
(1) I find that students sometimes use degrees Celsius mistakenly in equations when they should be using values of temperature in degrees Kelvin. This might problem might or might not go away by defining a new temperature scale.
(2) The units of J/mol/K  for molar entropy and specific heat seems to have no intuitive analogy. I think that part of the reason that entropy is such a hard concept for people to grasp is that we in the engineering community tend to use units that are non-intuitive.  (Energy per mole per temperature)
(3) Some students seem to confuse voltage with voltage difference, and I think that this is in part due to the fact that we are constantly using different zero points for voltage.

So, what else can we do to make thermodynamics easier to teach and to apply? Here's an attempt to formulate thermodynamics in Earth-centric units.

(1) We can change the temperature scale, such that the temperature of the Earth's environment is approximately equal to 1, while keeping the absolute minimum temperature of 0. Temperature would go from 0 [AT]  (same as 0 Kelvin)  to infinite [AT]  (same as infinite Kelvin), and in between there's T = 1 [AT]  which corresponds to T = 300 [K],   where [AT] is the unit for Earth's average atmospheric temperature. A substance that is hotter than the Earth's average temperature will be greater than 1 [AT] and a substance that is colder than the Earth's average temperature will be less than 1 [AT].

(2) We can change the units of entropy such that the units are dimensionless (as is done in statistical mechanical courses.) The reason, of course, that entropy should be dimensionless is that entropy is the logarithm of the microstates of a given macrostate, which is the logarithm of the number of exchange symmetries between similar particles. Entropy is the logarithm of a countable number, and hence, we should be assigning it dimensionless units, even in the engineering community. We still need to have a constant out front in the Boltzmann equation {i.e. Entropy = constant * ln (microstates)} because the natural logarithm of the number of exchange symmetries in a mole of a gas is still extremely large, but the constant out front will be dimensionless.

(3) Since entropy should be dimensionless and since temperature is defined as the derivative of the energy in a system with respect to the entropy in a system, then temperature should have the same dimensions as the energy (as is done in stat mech textbooks.) The energy of a substance will have units of [AT]. There will be a fixed conversion factors between [AT],  [J] and [kWh]. This means, of course, that specific heats will have dimensions of inverse moles, [mol-1].  The molar internal energy of an ideal gas is the integral of the specific heat at constant volume times dT (the differential temperature). This yields a dimensionally correct statement when the specific heat has units of [mol-1]. While this is correct dimensionally, it would be convenient if the value of the specific heat at constant volume were equal to 1/2 times the number of degrees of freedom times a constant...for example, the specific heat at constant volume should be equal to 3/2 C for an ideal monatomic gas and 5/2 C for an ideal diatomic gas (that is when the temperature is well below the energy of the n=1 vibrational mode.) Note that since temperature and energy could be stated using the same units, then this last statement is dimensionally correct. To do this, the constant that relates the 1/2 times the degrees of freedom to the specific heat at constant volume should be equal to 1 [1/collection] times a Constant. This means that we have to define a quantity of atoms/molecules such that the collection of ideal atoms has an internal energy equal to the number of degrees of freedom times the temperature. "Collection" is like "mole" but it could be a different number of atoms/molecules, such that the thermodynamics calculations are easier to do.

A "Collection of atoms" is equal to a "mole of atoms" time the ratio of [AT]/[J] divided by 2494.

(4) Now, we get to the kicker. Since we defined the Earth's temperature to be equal to 1 [AT], then the exergy destruction is equal to the irreversible entropy generation [hr-1] time the Earth's temperature 1 [AT]. Exergy destruction can then we converted to [kW] by converting [AT/hr] into [kW], using the fixed conversion factor between [AT] and [kWh].

(5) Note that we have already defined an earth-centric unit for pressure [atm], and this is already used in the thermodynamics literature, and it has already been used as a reference point when calculating the Gibbs free energy of substances. The zero point of pressure (like temperature) is not arbitrary, and for both MPa scale and atm scale, the zero point of pressure is the same. So, there's no real difficulty in moving from the MPa scale to the atm scale, as long as you have the correct conversion factors.

(6) We can define the zero point of voltage to be the voltage of a platinum electrode in equilibrium with the Earth's environment. While this choice of zero point voltage is somewhat arbitrary, it's not arbitrary for fuel cell system engineers. The cathode of PEM fuel cells systems is normally a Pt-electrode in contact with air and water. A platinum electrode at room temperature in contact with 1 atm water fully saturated with room air could be used as the zero point voltage. (Of course, the reason that most people don't use this as the zero point voltage is that it's often easier and more precise to use a sealed liquid mercury reference electrode in contact with 1 atm hydrogen. The hydrogen oxidation kinetics are a lot faster than the oxygen reduction kinetics, so the reference electrode can be a lot smaller if you need to measure voltages of systems that are changing rapidly with time. Also, you can always look up the conversion between the voltage of a mercury electrode in contact with hydrogen and the voltage of a Pt-electrode in contact with air. For information on how to go between different zero point voltages, check out this linked presentation or check out the wiki site on electrochemistry.)

However, it would be nice if we set the zero point of voltage equal to the Pt-electrode in contact with air, because then we could place a Pt-electrode into a different substance and measure the voltage with respect to the voltage in air. This would be related to the molar exergy of the substance times the number of electrons generated at the electrode per mole of the substance. So, let's say that we placed an electrode into a pressurized natural gas tank. We would measure a certain voltage difference, and this would be a way of measuring its molar exergy per z (where z is the number of electrons produced at an electrode for each molecule of the substance), and we could relate this back to units of [AT/C], where C= Coulomb, or to [kWh/mol e-].
The molar exergy would be equal to the absolute value of voltage times z. (Note that voltage could be positive or negative with respect to the Pt-electrode in room air, but the exergy would always be positive because you can generate work from any substance not-in-equilibrium with the Earth's environment if the Earth's environment is the thermal/mechanical/chemical sink. The direction of electricity would be different if the voltage were positive or negative, but you could switch the direction of your load, and still power your device.)
Things could get a little tricky here because what if we place the non-reference electrode into the middle of a carbon dioxide pipeline. We would measure a voltage because the substance is not-in-equilibrium with the Earth's environment, but the calculation of the exergy might be tricky because it's not obvious what value we should use for z for carbon dioxide.

So, I'd like to conclude this post by summarizing what I've stated above and noting again that this set of Earth-centric units for temperature and voltage is completely arbitrary, but it might make teaching thermodynamics easier. Here's the summary:
(1) We can arbitrary define the Earth's average temperature to have a value of approximate 1 [AT]
(2) Entropy should be a non-dimensional quantity  [-]
(3) Energy will then have the same units as temperature, and specific heat will then have dimensions of inverse number [mol-1]
(4) Zero point voltage could be defined as a Pt-electrode in thermal, mechanical, and chemical equilibrium with the Earth's environment.   (While the Earth's environment is not constant, we have already defined the standard state to be roughly 25degC, 1 atm, 76% N2, 20.5% O2, ~2% H2O, 1%Ar, 400ppm CO2, etc... We could use the same standard conditions in the definition of the zero point voltage.)
(5) I'll conclude by discussing how to go beyond thermodynamics, and to relate this set of units to the units used by economists. If we were to live in a society in which the average price of purchasing useful work were held constant, then there would be a constant average conversion between units of [kWh] and [\$]. This wouldn't mean that the price of electricity would always remain the same. It's just that now [\$], [AT], [kWh], and [J] all are in dimensions of energy/work and there's a conversion between the []'s. In the case of [\$]'s, the conversion is only an average conversion factor. But if on average were \$1 buys you on average 20 kWh of electricity, then if you are paying \$2 for 20 kWh, then you know that you are buying electricity that took for work to generate than the average amount of work required to generate 20 kWh. If you only have to spend \$0.50 for 20 kWh of electricity, then you know that less work went into generating that electricity than the average amount of useful work required to generate electricity.

Currency, energy, work, exergy, temperature are not the same things, but they have the same fundamental dimensions of mass multiplied by length squared divided time squared. Thermo-economics should reflect this similarity. Growth rates and irreversible entropy generation rates have dimensions of inverse time, such as [1/yr], and thermo-economics should also reflect this similarity.