## First Law of Thermodynamics – II

This is the second half of the notes I’ve taken for myself over the years concerning the first law of thermodynamics.  See the previous post for why this is appearing.  Even the cognoscenti might find something of interest in this post — e,g. an explanation of why heat capacity decreases with decreasing temperature, using the fluctuation dissipation theorem. Anecdote aficionados might like to hear about the Hawking Preskill bet.

First Law of ThermodynamicsThe energy of an isolated system (no heat in or out, no work done on or by) is constant

Atkins3 p. 48 Heat capacity is measured per mole, and is the amount of heat (measured how?) raising the temperature of a substance (at a given temperature) by a certain amount.   The heat capacity concept was developed by James Black in 1760. Interestingly, it led to the demise of the idea that heat was a fluid.  He heated water and mercury over the same flame, and noted that mercury got hotter.   The heat absorbed or evolved by the system can be monitored by noting the temperature changes taking place in a surrounding bath (with a known volume of substance and a known heat capacity — this again begs the question of how heat is actually measured — perhaps by using the mechanical equivalent of heat, and then using this to determine heat capacity).

Atkins7 p. 44 — Actually heat capacity can be measured either per mole (molar heat capacity) or per gram (specific heat capacity)

Atkins7 p. 44 — In general, heat capacities depend on the temperature and decrease at low temperatures (see later for the explanation which involves the fluctuation dissipation theorem)

At the temperature of a phase transition the heat capacity of a sample is infinite (heat just keeps being added to ice to melt it, but the temperature of the ice and/or the water doesn’t change, so deltaT is zero and Cv=heat supplied at constant volume/deltaT.

A very nice discussion of exact and inexact differentials as they apply to internal energy and work respectively is found in PChem McQuarrie & Simon p. 773. — which refers to p. 688 and before — p. 688 shows a differential which is not exact –.e.g. inexact.

In the case of P = f(T,V)  dp = dP/dT(v) dT + dP/dV(t)dV

is the form of the exact differential.  Suppose you had an expression for dP/dT(v) and another for dP/dV(t) — never mind how you get them.   Differentiate the first with respect to V and the second with respect to T, and you should get the same thing.  This is because the order of partial differentiation of the two variables won’t matter if p = f(T, V).

If a thermodynamic property is an exact differential of some other thermodynamic properties it is a state function because the differential is exact, and integration is independent of path.   Examples of state functions include enthalpy and internal energy. Atkins3 p. 59 also goes over these points.   Atkins uses deltaX for state functions, and deltaBarX (dBarX) for path dependent functions.

Work and heat depend on the path taken between two points, so they are never exact differentials (state functions).  Work cannot be defined as a function of variables determining the thermodynamic state of the system (e.g.  state variables).  Recall — here state means equilibrium state.

Atkins p. 49 — Isochoric heat capacity (Cv) is the amount of heat to cause a change in temperature at constant volume of the substance.  Isobaric heat capacity  (Cp) is the amount of heat to cause a change in temperature at constant pressure.   Isobaric gains and losses of heat aren’t always reflected in the temperatures of the initial and final states, as there may have been expansion or contraction allowing for the performance of work.  Remember that pressure * volume change == work.

Since the first law can be wriitten as dU = dq + dw — change in internal energy = change in heat + change in work.  At constant volume — no PV work can be done, so dU = dq (assuming that no other type of work such as electrical work is done).

Thus Cv = (dU/dT)v  Cv is the heat capacity at constant volume.

Thus from a physical measurement (the heat capacity at constant volume) we can measure the change in internal energy (a state function).   Assume we can measure temperature.  We can measure the mechanical equivalent of heat by measuring how much stirring  it takes to raise the temperature of a mole of a substance by a certain amount.  We can measure the heat capacity of a substance (at a given temperature), and thus know by how much we are changing the internal energy (thanks to the first law of thermodynamics)

Atkins3 p. 50  Enthalpy — denoted H — it is the internal energy plus pressure * volume.  H = U + PV.   Since U, P, V are state functions, so is H.  Note that the change in enthalpy of a reaction or a physical process doesn’t depend on the work done by or onto the system.  Since H is a state function, it is path independent.

The enthalpy change of a reaction (see below) differs from the internal energy change of a reaction because at constant pressure (e.g. that of a reaction in an open vessel) there is a difference in volume between reactants and products — this means that in the course of the reaction some pV work is done (either on the system or by the system).   HOWEVER IN REACTIONS INVOLVING ONLY SOLIDS OR LIQUIDS THE VOLUMES OF THE PRODUCTS AND REACTANTS ARE ABOUT THE SAME, AND EXCEPT UNDER SOME GEOPHYSICAL CONDITIONS WHERE PRESSURES ARE LARGE, THE CHANGE IN ENTHALPY IN A GIVEN REACTION IS THE SAME AS THE CHANGE IN INTERNAL ENERGY (FOR LIQUIDS AND SOLIDS ONLY).   No reaction, however extreme is going to change atmospheric pressure (which is where most reactions are carried out).

This is why chemists love enthalpy — most reactions are done in open vessels with constant (atmospheric) pressure, so the heat energy absorbed or produced (which is easy to measure) allows them to measure the enthalpy directly — e.g.  q = deltaH  — this is why the enthalpy change of a reaction is sometimes called the heat of reaction. In what follows, the rather weird looking || on one line with the v immediately underneath in the next line is supposed to stand for ==> (implies).

H = U + PV   <==>   U = H – PV.
||   ; Assuming constant P (so VdP == O ) and that only PV
work is done on or by the system
deltaU = deltaH – PdeltaV
||   ;  PdeltaV == deltaW
v
deltaU = deltaQ + deltaW  ; first law
||
v
deltaH = deltaQ     ; assuming constant P and  assuming that
; only PV work (and not much of that) is

; done on/by the system

So the heat capacity at constant pressure (Cp) is the ratio (at the limit) of heat to temperature change (deltaT) as deltaT goes to zero

deltaQ = Cp * deltaT

So given that deltaH = deltaQ at constant pressure (when only PV work done)

deltaH/deltaT at constant pressure =  Cp

Atkins 4Laws p. 38  The energy released as heat by a system free to expand or contract as a process occurs, as distinct from the total energy released in the same process, is exactly equal to the change in enthalpy of the system (provided the system is free to expand in an atmosphere that exerts a constant pressure on the system).  What’s the difference?  Where does the extra internal energy go (or come from)? It goes into the work of expanding (or contracting) volume against a constant pressure.  One can get this extra work out as heat if the reaction is done in a closed container which can’t expand.

To change a liquid into vapor requires energy to separate the molecules from each other.  This is supplied as heat — e.g. making use of a temperature difference between the liquid and its surrounding.   The extra energy of the vapor was called ‘latent heatl’ because it was released when the vapor condensed to a liquid, and was in some sense latent in the vapor.   Latent heat has been replaced by the term enthalpy of vaporization.

Atkins 4Laws p. 42 — The fluctuation dissipation theorem says that the ability of a system to dissipate (essentially absorb) energy is proportional to the magnitudes of the fluctuations about its mean value in a corresponding property (whatever that is).  Heat capacity is a dissipation term: it is a measure of the ability of a substance to absorb energy supplied to it as heat.   The corresponding fluctuation term is the spread of a population over the energy states of the system — when all the molecules are in a single state (say near or at absolute zero) there is no spread of populations, so the heat capacity of the system is zero.  In most cases the spread of populations increases with increasing temeperature, so heat capacity typically increases with rising temperature.

Here’s what I have about the theorem  from elsewhere  –[ Nature vol. 431 pp. 28 – 29 ’04 ] From Einstein in his annus mirabilis (1905).  The fluctuations in classical Brownian motion which make a pollen particle jitter, also cause friction if the particle is dragged through the medium in which it is embedded (water).  The fluctuation of the particle at rest has the same origin as the dissipation of the motion of a moving particle subject to an external force.   It is one of the deepest results of thermodynamics and statistical physics.

The heat capacity at constant pressure (and no other types of work such as electrical being done on or produced by the system) is thus defined as dH/dT == Cp — the volume doesn’t change –.  H plays a central role in chemistry because we are so often concerned with processes occurring at constant pressure (reactions, occurring in open vessels, the body etc. etc.).    P is taken to be the pressure of the system, the form pV is a part of the general definition of H for any system, and does not imply a restriction to perfect gases.

For a perfect gas H = U  + nRT (as H = U + PV and PV = nRT)

Atkins7 p. 46 — When a system is subjected to a constant pressure, and only expansion work can occur, the change in enthalpy is equal to the energy supplied as heat.   Heating liquid water by an electric coil doesn’t expand its volume by much (although it does to some extent) so most of the energy going into the enthalpy change goes to the internal energy of the water (but all of it goes to the enthalpy).

The thermodynmic internal energy is divided per molecule into several forms.
l. The energy stored in molecular bonds
2. The energy of molecular translation
3.  The energy of rotations of the molecule

4.  The energy of vibration of the various bonds of the molecule.

Thus we can’t take the average energy of a molecule at a given temperature and determine how fast it is moving, if the molecule can rotate or vibrate.  How these energies are partitioned isn’t clear to me.

For a substance made up of molecules one must add the energy of interaction of the molecules.
Heating is the transfer of energy as a result of vigorous random molecular motion into the surroundings.
Work involves organized motion — when a weight is raised or lowered, its particles move in an organized way, not just chaotically (although they do move chaotically as well).
Work is identified as energy transfer making use of the coherent motion of particles in the surroundings, and heat is energy transfer making use of their random thermal motion.

The distinction between work and heat must be made in the surroundings (not within the system).

Atkins3 p. 57 Extensive properties depend on the amount of a substance present.  Examples include internal energy, mass, volume, heat capacity (not per mole).  Extensive properties are also known as colligative properties.  Intensive properties depend on the state of a substance and don’t depend on the amount present — examples include temperature, molar heat capacity (absorbable heat per mole), pressure, density, viscosity, concentration — actually any molar property (a property per mole).  Ultimately, when carried to extremely small amounts of the substance, intensive properties disappear — what is the density or pressure of a single molecule.  More to the point — what is the concentration of a molecule in a very small volume (where none is likely to be found)?

Some properties depend only on the present state of the system and not how it was prepared.  Examples include internal energy, enthalpy, volume, pressure and most importantly, temperature.  These are called state functions Other properties depend on how the state was prepared.  These are called path functions.

Internal energy is a state function (according to the first law by some contorted reasoning).  If it were a path function, one could cycle between two states of a system by two paths (one emitting work to the outside system, and the other not emitting work) and get a perpetual motion machine.

A reaction vessel is a thermodynamic system.  The energy transferred as heat during a reaction is called q.  q is a path function and not a state function.  Therefore q depends on HOW the reaction is carried out rather than on the reaction itself.    It is better to discuss energy changes during the course of a reaction using state functions — these don’t depend on how the reaction is carried out (e.g. reversibly or not).

If energy is transferred as heat at constant volume, AND if no other kind of work is done (what other kinds?  electrical? mechanical? — how many kinds are there? ) then the change of internal energy is equal (in absolute amount) to the heat transferred (in absolute amount).

[ Science vol. 305 p. 586 ’04 ] Just as scientists in the 19th century figured out that energy can neither be created nor destroyed, many 20th century physicists concluded that information is also conserved.    Black holes posed a big exception to this as information (as well as light or mass) never gets out.   Hawking said it would lose information, Caltech’s Preskill said that information would be safe until the black hole disgorged it.

Hawking recently proved to his sattisfaction using the Euclidean path integral method, that information isn’t destroyed when it falls into a black hole.   This implies that black holes aren’t portals to another universe.  Hawking gave Preskill a Baseball Encyclopedia.

Reactions for which deltaH > 0 are called ENDOTHERMIC — enthalpy being added to the system, while those in which deltaH < 0 are called EXOTHERMIC.    Thus emission of heat by a reaction loses internal energy (and enthalpy too)  for the system which is why reactions giving off heat have negative deltaH ! It all fits with the sign convention that work done on or heat added to the system increases the internal energy of the system.  So heat being given off lowers the energy of the system.  Internal energy is like height.  Reactions go from higher internal energy to low.

Enthalpies of reactions are reported for reactants and products in their standard states.  The standard state is most stable form of an element at a given temperature (usually 25 C == 298.15) and always 1 bar (100,000 pascals)  — Atmospheric pressure is 101325 Pascals — recall that a Pascal is 1 Newton/sq. meter.   For an element in its standard state the enthalpy of formation of this state is taken as zero.  For carbon, the standard state is graphite.  Hydrogen, oxygen and nitrogen are biatomic gases in their standard states.

The standard enthalpy of formation of a compound is always relative to the elements making up the compound in their reference phases — the thermodynamically most stable phase under standard conditions (except for phosphorus — where white phosphorus is taken as the reference phase).    The standard enthalpy of formation of any element in its reference state is taken to be zero.

The reaction is considered to begin with the reactants in an unmixed state (there is such a thing as the enthalpy of mixing) and the products in an unmixed state.  In the case of ionic reactions in solution, the enthalpy changes occompanying mixing and unmixing are insignificant in comparison with the contribution from the reaction itself.

When considering the enthalpy change of a reaction, one subtracts the enthalpies of formation of the reactants from the enthalpies of formation of the products.  You must multiply the enthalpy of formation of each moiety by the number of moles of each moiety in the reaction.

Hess’s law — you can add the enthalpies of each of a sequence of reactions together.   This is because enthalpy is a state function (as is deltaH for that reason).

The change in internal energy for a reaction (deltaU) is equivalent to the heat emitted or absorbed if the products and reactants have the same volume (so no work is done).  The change in enthalpy for a reaction (deltaH) is equivalent to the heat emitted or absorbed at constant PRESSURE (assuming only PV work is done to or by the reaction).   However, since the volumes of reactants and products are essentially the same in solids and liquids, the volume doesn’t change significantly, so both the enthalpy and internal energy change of a reaction is the same in solution.

There is a description of the adiabatic bomb calorimeter on Atkins3 p. 86.   The calorimeter is immersed in an external water bath (and contains an internal water bath — inside bath).  The temperature of the external bath is adjusted to that of the internal water bath  so no heat exchange occurs between the  internal bath of the calorimeter and the external bath (e. g. the process is adiabatic (no heat energy is exchanged) when the calorimeter is considered as ‘the system’).  The temperature of the bath inside the calorimeter certainly rises or falls as the reaction takes place.  The reason it is called a bomb calorimeter is because the walls of the bomb in the internal bath are quite sturdy so that no matter how much energy is released, the volume doesn’t change and so no work is done.   The device can be used for combustion studies as well.   One measures the change in temperature of the bath inside the calorimeter and finds the amount of heat of the reaction using the heat capacity (and the volume of the internal bath) of water at constant volume.   To be truly accurate the dependence of the heat capacity of water on temperature should also be known.

Atkins3 p. 87  Varieties of enthalpy.   The enthalpy of sublimation of carbon (from graphite) was quite hard to neasure, but quite important for organic chemistry.   It is 716.68 kiloJoules/mole.  Sublimation is just one type of phase transition.  Other phase transitions have their own enthalpies — these include enthalpy of vaporization (how in the world can volume be constant? — it doesn’t have to be — but pressure does ! ), melting (latent heat of melting.

The enthalpy of solution is usually measured as what happens at infinite dilution (so the molecules of what is being dissolved don’t interact with each other).   The enthalpy of formation of a compound in solution is the addition of the enthalpy of formation of the substance by itself added to the enthalpy of solution of the compound (by Hess’s law).   Enthalpies can be added together because they are state functions.    For reasons that aren’t entirely clear, the enthalpy of formation of hydrogen ion in water is taken to be zero at all temperatures.

As an example of how complicated things can get — consider the enthalpy of formation of NaCl in solution.
l. Start with the enthalpy of sublimation of sodium solid (reference state at 25 C) into a gas.
2. Next the enthapy of ionization of sodium gas into Na+ and an electron.  This value is obtained by spectroscopy.  Volume changes should be added in here, but volume changes cancel when the electron joins a chlorine atom.
3. Next the bond dissociation enthalpy of Cl2 (again the reference state of chlorine at 25 C) into Cl atoms
4. Next the bond association enthalpy of Cl atom + electron to chloride ion in the gasseous state.  Some values are from spectroscopy and others are from calculation.  Here is where the volume change is cancelled out.
5. Next the enthalpy of formation of NaCl as a solid from a gas is added in.  This is lattice enthalpy.  At this point we have gone from Na (solid) and Cl2 (gas) at 25 C to NaCl as a solid.

6. Last the enthalpy of solution of NaCl in solvent (usually water) is measured.

Atkins7 p. 46 — If a process involves only solids or liquids, the values of changes in enthalpy (deltaH) and internal energy (deltaU) are ‘almost identical’.  To really measure deltaH — Atkins talks about a thermally insulated vessel open to the atmosphere (so how can it be thermally insulated — unless air is such a poor conductor of heat < which it is ! >)  — this is the isobaric calorimeter.

Then they talk about a differential scanning calorimeter — which measures the heat transferred to or from a sample at constant pressure during a physical or chemical change.

Atkins7 p. 52 — What happens when a gas expands adiabatically  — no heat enters or leaves  the gas.  Assume the expansion is reversible (so external pressure and internal are the same).  I just don’t follow the discussion.    Even so the temperature of the gas drops — because the gas is doing work against external pressure — what if it is expanding into a vacuum ?

Thermochemistry — the study of the heat produced or absorbed by chemical reactions.  This is a branch of thermodynamics because a reaction vessel and its contents form a system resulting in the exchange of energy between the system and the surroundings.    Calorimetry can be used to measure the heat absorbed or produced by a reaction.

If the reaction occurs at constant volume — heat in or out measures the change in U.  If it occurs at constant pressure — heat in or out measures the change in enthalpy (H) — assuming no other type of work (electrical, expansion, surface expansion, extension  — Table 2.1 p. 39 Atkins7) is done on or by the system.

Enthalpies can be reported for physical changes (with no chemical reaction occuring) — such as vaporization, freezing, sublimation.   Quick — when you melt something is the enthalpy positive or negative.  You have to supply heat from outside, so internal enthalpy (and energy) are raised so the enthalpy of melting, vaporization, sublimation is positive.

Since enthalpy is a state function, it is independent of the path that brings a system to that state, so we can put a solid into a gas by (1) sublimation or by (2) melting followed by evaporation — the enthalpy change of processes (1) and (2) will be the same.    Another consequence of enthalpy being a state function is that the enthalpies of a process going one way will be the negative of the process going the other way.   This is true for both physical and chemical changes.

You can write chemical reactions as occuring between compounds in their pure states
(e.g. methane + 2 O2 –> CO2 + 2H20 + heat given off < negative H > )
The enthalpies of mixing methane with O2 and then separating the products can be ignored, because they largely cancel — however, not for ionic reactions in solution.

You must multiply the enthalpy of formation per mole of the reactants by the number of moles involved in the reaction.

Atkins7 p. 59 — Another enthalpy is the specific enthalpy — the enthalpy of combustion per gram of material (e.g. like specific gravity).  Useful in biochemistry — where substances are rarely pure — one can talk about the specific enthalpy of combustion of a gram of fat.   One can also talk about the enthalpy density — the enthalpy of combustion of a liter of material.

Atkins7 p. 62 — Computer aided molecular modeling is becoming the technique of choice for estimating standard enthalpies of formation of molecules with complex 3 dimensional structures.  The idea of finding the enthalpy of formation of a group (such as a methyl group) is regarded as primitive and old fashioned — in any enert such modeling is derived from a series of compounds.

The modeling predicts the relative stabilities of conformational isomers fairly well — ‘good agreement between calculated and experimental values is relatively rare.’

Heat added or lost is always an inexact differential (it depends on the path taken between initial and final states).  It is zero for an adiabatic process (even though the temperature may change during one).

The internal energy of a perfect gas is independent of volume at constant temperature (because its internal energy arises only from the kinetic energy of its molecules).  For any isothermal change in a gas, its energy doesn’t change.   However, it isn’t zero for a nonperfect gas (which is all gases).  Joule tried to measure the internal energy (dU/dV)T by letting a gas at 22 atmospheres expand into a vacuum.   He got zero, but his apparatus was extremely inaccurate.  As the volume which a gas is confined in shrinks, if the molecules of gas attract each other, the internal energy (U) will diminish, while it will increase if the molecules repel each other.  Enough squishing and you get a liquid and then a solid.