First Law of Thermodynamics – I

The second chapter of Anslyn & Doughterty heavily involves thermodynamics.  Organic chemists get by with a basically qualitative understanding of the subject, but it’s always good to go deeper.  So I went to the notes I’ve taken on the subject over the years before plunging into Ch. 2.  I found them useful and will post the set of them here in the hopes that some of the readers will as well.  Back in the day computer memory was stored on iron cores (not silicon), programming was done in assembly language (or even worse, machine language) and errors were frequent (they still are) but much harder to find.  Computers crashed with dismaying regularity, and the only thing to do was look at the state of the computers memory to find out what had gone wrong — this became known as a core dump.  This series of posts is basically a core dump of the notes I’ve taken on thermodynamics over the years for my one benefit and understanding.

Be warned.  The notes are jumbled up and repetitive to some extent. If I don’t understand something, I say so  They certainly aren’t the way to learn the subject.  But if you’ve gone through thermodynamics in the past and haven’t looked at it for a while, they might be a good way to get up to speed.  Hopefully some of you will find them useful.

Sources are the various editions of Atkins book (Atkins3 is the third edition).  Berry is the big text he wrote on PChem.  McQuarrie and Simon should be familiar to most.  Atkins 4Laws is a small book (4 Laws that Drive the Universe) he wrote with very little math in it explaining what thermodynamics really is about.  I liked it a lot.

       The first law of thermodynamics: The energy of an isolated system (no heat in or out, no work done on or by) is constant

       The name thermodynamics reflects its origin in the steam engine (Invented by James Watt in 1780) etc. etc. and an interest in turning heat into motion.  Stowe “An introduction to thermodynamics and statistical mechanics” p. 4 — 2nd Ed. 2007.  This is why the original sign convention for work done by the system had it positive (rather than negative as work is defined today).  Heat added to the system has always been positive. 

        Presently people are interested in direct processes — where energy is converted from one form to another, without much heat being involved — sunlight into electricity, chemical energy into electrical energy (fuel cells, batteries).  

        Atkins 4Laws p. 45 — The first law is essentially based on the conservation of energy, the fact that energy can neither be created nor destroyed.  Noether’s theorem says that every conservation law corresponds to a symmetry.  In the case of the conservation of energy, the symmetry is that of the shape of time.  Energy is conserved because time is uniform — time flows steadily, it doesn’t bunch up and run faster than spread out and run slowly.   If it did energy wouldn’t be conserved. 

        Gribbbin — “Deep Simplicity” p. 110 — “In a sense, classical thermodynamics pretends that time does not exist. Systems are described in terms of infinitesimally small changes that would take an infinite amount of time to shift from one state to another.”

       The first law assumes that the system under consideration has no energy flowing through it and that the system does no work e.g. the system is closed.   It is in closed systems that we encounter time reversibility and Poincare recurrences; in open systems (in which energy is flowing through the system) we encounter irreversibility and an arrow of time. 

        Atkins7 p. 35 — The internal energy of an isolated system is constant.  How internal energy is measured isn’t given.  Work done ON a system, heat transferred TO a system raises the internal energy.  Berry PChem2 p. 371 — Heat is simply a term for energy which crosses the boundary of a closed (no mass in or out) in a form other than that of work.  “Heat is just energy in transit to or from the system”

      U stands for internal energy (but U must be abbreviating some word — probably in German) deltaU = q + w  (q is heat, w is work).  However this formulation also says something else — work and heat are equivalent forms of energy.  (This simple law took a huge amount of work to really establish).  This is the acquisitive sign convention — w > 0, q > 0 if energy is transferred to the system as work or heat.   We are viewing the flow of energy as work or heat from the system’s perspective.   This is the way Tinoco’s PChem book and the course I took regard things.

       Atkins7 p. 36 — a nice derivation of the first law assuming that all we know is how to measure is work (in the physicist’s sense of force x distance).  Consider an adiabatic system (no heat in or out — this assumes we know what heat is and how to insulate for it). — and also that we know how to measure temperature).  Experimentally, it was found that the same increase in temperature in this system is brought about by the same quantity of ANY kind of work done on the system (which we do know how to measure but recall that finding the mechanical equivalent of heat by Joule was a very big deal).   Also measuring temperature was a big deal.   This also assumes that we know how to measure pressure, volume and temperature (or any other state variable).  

        Berry PChem2 p. 379 The only way a thermodynamic state can be changed in a system with adiabatic walls is through work. 

       Measuring work by passing an electric current seems rather hairy, measuring the work done by stirring a solution does not.   However, the work done by an electric motor (using current) lifting a weight against gravity is clear, so the work of a current flow can easily be translated into the work of lifting an object. 

       The restatement of the first law says that the work needed to change an adiabetic system from one specified state to another specified state (without saying just what it takes to specify the state) is the same however the work is done.   This is like climbing a mountain, the altitude at the begining and end of a path doesn’t depend on the route taken.  The altitude is independent of the path.   The observation that a state is independent of the path implies the existence of a state function.   So altitude and energy are both state functions.

       How to measure heat?  It is the difference between two changes of state under different conditions (adiabatic and diathermic)  e.g. between adiabatic work (no heat transfer) and diathermic work (heat transferred) — again one measures the ‘state’ of a system, but Atkins7 gives no clue (at this point) about how such things are done.  We are to infer that pressure volume, temperature, and number of moles are all you need for a gas (from the previous discussion). 

       However, this does give one a mechanical definition of heat in terms of work. 

       Atkins3 p. 38 Chapter 2  Amazingly, he never defines heat in this chapter.  Heat is what is transferred between objects at different temperatures allowed to come to (thermal?) equilibrium with each other  This isn’t surprising as chemists thought heat was some type of fluid, and it wasn’t until the mid 1800s that physicists said that heat was some type of motion, without knowing just what heat was a motion of (since the atomic constitution of matter remained controversial even into the early 20th century).  

       Atkins (Four Laws) p. 30 “In thermodynamics, heat is not an entity or even a form of energy, heat is a mode of transfer of energy”  — heat is not a fluid rather —  “heat is the name of a process, not the name of an entity” — heat is the transfer of energy as the result of a temperature difference.

        Berry (PChem2 p. 371) Heat is a term for energy which crosses the boundary of a closed system in a form other than that of work. 

       Recall that we have been able to define the internal energy of a system as a state function.  Atkins Four Laws p. 28 “The amount of energy that is transferred as heat into or out of the system can be measured very simply” FIRST — we measure the work required to bring about a given change in the adiabatic system (no heat transferred), and then SECOND — we measure the work required to bring about the same change of state in the diathermic system, and take the difference — the difference is the energy transferred as heat.  A point to note is that the measurement of the rather elusive concept of ‘heat’ has been put on a purely mechanical foundation as the difference in the heights through which a weight falls to bring about a given change of state under two different conditions”  — e.g. adiabatic and diathermic. 

       On an atomic level, the difference between work and heat is quite clear.  Work performed by a system on its surroundings is the transfer of energy which causes a uniform motion of atoms (e.g. the motion of an aggregate of atoms) in the surroundings — e.g. the lifting of a weight against gravity.  Heat is a transfer of energy to the surroundings but in the form of increased random motion of the atoms in the surroundings.

       Once energy is transferred INTO a system, either (1) by making use of the uniform motion of atoms in the surroundings (a falling weight) — which could be used to turn a rotor inside the system increasing its temperature, or (2) by causing the transfer of energy as heat, there is no memory of how it was transferred.  Atkins 4Laws p. 33 — he does make a distinction between how energy is stored — e.g. kinetic energy of the atoms or the potential energy due to the position of the system.  This energy can be withdrawn either as heat or as work.

       One statement of the first law is “the internal energy of an isolated system is constant. 

      Further notes on the first law are gleaned from McQuarrie and Simon and are interspersed with those of Atkins3 and Atkins7.

      Atkins7 p. 33 — The internal energy of  a system is called its energy (U).  It is the total kinetic and potential energy of the molecules in the system under consideration.  (2 Aug ’04 — What about vibration? potential energy relative to what? )

      Atkins7 p. 31 — the energy of a system is its capacity to do work.   A boundary between the system under observation and its surroundings permitting the transfer of heat is called diathermic.  A boundary not permitting heat transfer between the system and its surroundings is called adiabatic. 

      McQ p. 766 – We define heat q, to be the manner of energy  transfer that results from a temperature difference between the system and its surroundings.  Heat input to the system is considered positive as it raises the internal energy.   McQ defines work (w) to be the transfer of energy between the system of interest and its surroundings as a result of the existence of unbalanced forces between the two.   Work can always be related to the raising or lowering of a mass in the surroundings of the system. 

        When a gas expands against a constant external pressure, the work done by the gas doesn’t depend on how the pressure of the gas changes as it does the work — this is why work isn’t a state function.    The higher the pressure it expands against, the greater the work done by the gas.   Thus gas expanding to twice its size against two different external pressures will have done more work against the higher pressure.  The states of the gas at the beginning and end of both expansions are the same < 12/03 — couldn’t the temperature be different ? > , so work is not a state function.  Similar considerations apply to contraction of a gas.  

     Aha !  25 Dec ’01 — the reason the integrals of PV work of a gas and of the internal energy of a gas look so similar is that that you are integrating over state functions (at constant pressure its just over dV) and when integrating over dU, the internal energy depends only on the state of the gas.  In one case you subtract Vinitial from Vfinal and use Pexternal (at constant pressure), and in the other you subtract Ui from Uf.    P can be taken out of the integral because it is constant.  If P varied all over the place as the gas expanded, P would be some other function of volume and that function would stay inside the integral and have to be integrated yielding different amounts of work depending on the ‘path’ the pressure took (given by the function) as V varied from Vi to Vf.   This assumes that you understand path integrals which I don’t. 

     McQ p. 770 — If you compress a perfect gas so its temperature stays the same e.g. PiVi = PfVf, using external pressure just slightly greater than internal pressure, then P is a function of V — the process is reversible as well.  P = nRT/V, and integrating this over dV gives a logarithm — the point is that no extra work is done on the gas under these conditions.  When Pext is constant where does the extra work done on the gas go?    Does it heat the gas?   

      When a gas expands reversibly against a pressure infinitesmally lower than the pressure of the gas, PV work is done by the gas.  If heat doesn’t enter or leave the gas (adiabatic process) by giving up work the internal energy of the gas must drop.  This can only mean that the temperature of the gas drops in an adiabatic reversible expansion.  McQ notes that by a discussion in a previous chapter, the internal energy of an ideal gas depends only on the temperature ! !  Thus in an adiabatic reversible expansion the plot of P vs. V is not along an isotherm of PV = constant.   Heat can be added and subtracted reversibly (I thought the second law of thermodynamics prevented this).  Neither reversible work or reversible temperature change are state functions.  McQ p. 774. 
      Atkins3 starts with a truly dopey definition of work —  Work is done if the process could be used to bring about a change in the height of a weight somewhere in the surroundings.  

       Weight = acceleration of gravity * mass == force exerted on a mass.  Work is measured as force times distance (in this case as weight x height).  and it has the same units as energy (see  Joule).  

        Energy is the capacity to do work.  When work is done on an otherwise  isolated system its capacity to do work is increased (so its internal energy must be increased).   Heat can change the internal energy of a system.

       First law of thermodynamics — when a system changes from one state to another along any adiabatic pathway (no heat transfer into or out of the system) — the temperature of the system may well change in the process however,   see p. 86 where how this is measured with an adiabatic bomb calorimeter is at last described after 50 pages of confusion) the quantity of work done is the same irrespective of the means employed.   However, when heat is transferred all bets are off, as not all transferred heat can be used as work.  

       Atkins7 p. 33 —  Work is identified as energy transfer making use of the organized (e.g. correlated) motion of atoms in the surroundings.  Heat is identified as energy transfer making use of thermal motion in the surroundings.  

       This is analogous to noting that one’s vertical distance from the peak of a mountain is independent of the path of the ascent.  The independence of path of vertical distance implies the existence of a property of the mountain — which we call altitude.  

      The property of thermodynamic systems so expressed is called the internal energy of the system.    The first law of thermodynamics is that the internal energy of an isolated system is constant.   Work done ON the system is considered positive, as is heat added to the system. 

       Now let the system change between the same initial and final states as before, but allow heat to enter or leave the system.  The internal energy change is the same, but the amount of work between the two states might be different depending on the path chosen.    However, the sum of the work done and the heat exchanged add up to the change in internal energy.    The values of heat and work are dependent on the path taken.    

      deltaU (where U == internal energy) = work done on the system or – work done by the system + heat added to the system or – heat removed from the system

      If deltaU = 8, then 8 = 5+3, 6+2, 12-4, etc, etc.  

      The first law provides a way to define heat mechanically.  However, this requires a way measure the internal energy of a system — which isn’t described until the next chapter.  Here’s how to measure the mechanical equivalent of heat.  Assuming you know deltaU between two states, you measure the amount of work to go from one state to the other adiabatically (e.g. no heat transferred into and out of the system), and then measure the amount of heat required to go between the same two states doing no work.  If you then ASSUME that the first law is true, since work and heat are both energy, you have determined the mechanical equivalent of heat.   Joule in the mid 1800s was very involved in trying to measure the mechanical equivalent of heat.  

      1 Joule is the energy required to move 1 kilogram (2.2 pounds)  a distance of 1 meter (about 40 inches) EXACTLY THE SAME THING as the  work < force x distance >  required to move 1 kilogram (2.2 pounds) a distance of 1 meter (about 40 inches).   

      The Joule units are {kiloGram*(meter)2}/(second)2} — Note ! the same units as kinetic energy e.g. — mass * (velocity)^2.   

       The analogue in the CGS (centimeter gram second) system of units is the erg. 1 Joule is also .23 calories, and a calorie is the amount of energy (usually measured as heat) to raise 1 gram of water by 1 degree Centigrade between 14.5 to 15.5 degrees centigrade.  One calorie therefore is equivalent to 4.184 joules.  The calorie we talk about in food is really 1000 of these small calories.  So if you weigh 220 pounds (100 Kg) you have to walk 15 feet to burn 1 food calorie, or 15,000 feet to burn 1000 calories.  This is why it is so hard to lose weight. 

       The first law also implies that the internal energy of an isolated system can’t change — isolation means that no work is on or by the system, and no heat is transferred into or out of the system.    This leads to the impossibility of constructing a perpetual motion machine.    An isolated system however can undergo a change of state — liquid water below 32 C will freeze, but the energy of the system won’t change.  

        dw is the work done on a system, and dq is the heat supplied to the system.  They should be positive as both raise the internal energy of the system — which should be positive.    However w’ is the negative of w and is positive when the system does work on its surroundings.   w’ is what chemists usually measure.  

       Work is force x distance.   Interestingly, I’m battling (22 Dec ’01) with the idea of linear functionals and work is such an item — it takes a vector (force) and another (distance) and produces a scalar (work). 

       Pressure is force/area   so pressure x area = force, so pressure x area x distance equals work.  However, area x distance equals volume so pressure x volume = work.  This is why the gas constant R = PV/T is measured in Joules * Kelvin^-1 (Joules/Kelvin)

       weight = acceleration of gravity * mass == force exerted on a mass at the surface of the earth. 

       When a gas expands against a constant external pressure, the gas is doing work on the external environment, so the internal energy of the gas drops.   When a gas is compressed at constant external pressure its internal energy rises (work is being done on the gas).    Since PV =nRT the only way volume can decrease at constant (internal) pressure is if the temperature drops. However, this doesn’t happen at constant external pressure because the internal pressure rises and temperature stays the same (if no heat is transferred — e.g. the process being adiabatic).  

       Consider what happens when the internal pressure of a gas is higher than the external pressure on it.  The gas expands until the two pressures are equal, or until a mechanical stop is reached.   External pressure could be constant — e.g. it could be atmospheric.    The work done by the gas is then p(external) * deltaV.  The internal energy of the gas must drop by this much.   Assume no heat is transferred in the process.  All we know is initial and final volumes and the initial internal pressure and temperature.   We don’t know final pressure or final temperature.   If the process is adiabatic (no heat transferred), are we to assume that the final temperature is the same as the initial one.   We do know the internal energy has dropped by the amount of work done. 

       Atkins3 p. 46.  In thermodynamics, a reversible change is one that can be reversed by an infinitestimal modification of a variable.  If the expansion of a gas is to occur reversibly, we must ensure that at each stage the external pressure is only infinitesmially less than the internal pressure.   In some (poorly explained) way this involves the system being in equilibrium with its surroundings.  (12/03 — Atkins7 — still poorly explained).  Atkins7 p. 39 — A reversible change in thermodynamics is a change that can be reversed by an infinitestimal modification of a variable.  A system is in equilibrium with its surroundings if an infintesimal change in the conditions in OPPOSITE directions results in opposite changes in its state. 

      The reversible work done in expanding/contracting a gas is the integral of PdV.  P will change as volume changes — the pressure on the gas from without must (very nearly) equal the pressure of the gas at all times (so the process is reversible).  If the external pressure isn’t exactly the same as the gas, work produced won’t be reversible (which is why work isn’t a state function).  Very nice picture and explanation of this — Atkins7 p. 41 Fig 2.9 !

       Since work is the integral of pressure summed over infinitesmal volume increments, and since the pressure of a gas depends (by an equation of state) on volume and temperature, work done on or by a gas at constant temperature  depends only on volume — this is true whether the gas is perfect or not, as long as the pressure depends in some way only on volume and temperature (and we are keeping temperature constant).   This is why we need the equation of state to measure work  p = f(Vol, Temperature). If the temperature is kept constant (by being in contact with an external bath maintained at a constant temperature by the addition or subtraction of heat as necessary — see p. 86), than p depends only on V, and for a perfect gas p = nRT/V, so the integral involves evaluating a logarithm at initial and final volumes. 

       You get more work from a gas expanding isothermally and reversibly, than you do from a gas expanding at constant pressure, this is because to keep the gas at the same temperature as it expands, heat must be added to the gas (otherwise the temperature will drop), and all this heat is converted to work (because the expansion is reversible).  

       The MAXIMUM amount of work available from a system operating between specified initial and final states, and passing along a specified path is obtained when it is operating reversibly.   Unfortunately, in practice this normally means that the path has to be traversed infinitely slowly.  Consequently, reversible processes are called quasi-static 

        Berry PChem2 p. 375 — There are two types of thermodynamic variables intrinsic (P, T) and extensive (volume, mass).    Each intensive variable X has a conjugate extensive variable Y, the two being related by a particular work process.  Example:  the conjugate extensive variable of P is V, and P*V = energy (or work).

       The conjugate extensive variable Y of an intensive variable X is a quantity such that an infinitesmal change (dY) does an amount of work 

     e.g.  dBarWork = X * dY
                         Intensive variable    Extensive Variable
Fluid — or any
system which 
can expand and        Pressure              Volume
Surface film         Surface Tension        Area
Wire                       Tension                   Length
Capacitor               Potential                 Charge
Electrochemical      Electron motive       Charge
     cell                          force (emf)
Paramagnetic         Magnetic field           Magnetic
    solid                         strength (H)         moment

        In mechanics, generalized coordinates correspond to extensive variables and generalized forces correspond to intensive variables.  If the potential energy of a mechanical system is expressed as a function of several independent variables (the generalized coordinates), then the generalized force conjugate to a particular coordinate is defined as the negative of the partial derivative of the potential energy with respect to that coordinate with all the other coordinates held fixed.  

       The same thing happens in thermodynamics.  The intensive variable X conjugate to a given extensive variable Y is defined as 

         X = – dU/dY ;  partial derivative, all other extensive variables constant

     In both thermodynamics and mechanics, the product of the conjugate pair members must have the dimensions of energy. 

      The performance of a given type of work is always associated with a change in the corresponding extensive variable (in the system, the surroundings or both).   Whenever a boundary allows the performance of a particular kind of work, we say that the boundary transmits the conjugate intensive variable.  At equilibrium the intensive variable (P, T, mass) has the same value on both sides of the boundary (system, surroundings).   If the boundary doesn’t permit the performance of a particular type of work, than the intensive variable doesn’t have to have the same value on the two sides (even at equilibrium).  

        For a system initially at equilibrium to undergo a change of state, its boundaries must be capable of transmitting some intensive variable.  So either work or heat must pass across the boundary.  If no work is possible, than heat must flow.  
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