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(17) Understanding the State of Equilibrium · Mechanical Engineering Made Simple

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  Welcome back, design junkies, to mechanical engineering Made
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  simple. Today we're doing a deep dive
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  into some absolutely core to engineering analysis.
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  It's the state of equilibrium in thermodynamics.
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  That's right. You know most engineering design
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  pretty much everything from say, simple heat exchangers right up
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  to complex phase change systems. It all hinges on this
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  assumption, the assumption of equilibrium thermodynamics.
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  It basically defines the best possible outcome, you know, the
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  ideal state. But, and this is the kicker, the
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  real world is, well, it's messy. It's always running away from
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  this ideal state. And that's where it gets really
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  interesting, doesn't it? The actual effect of making this
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  ideal assumption because a lot of crucial processes, think
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  combustion for example, they happen so fast, way too fast to
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  actually hit that ideal stable state.
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  Exactly. And that's the context here
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  really. It's about the consequences of
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  not reaching equilibrium. Things like pollutants, carbon
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  monoxide, texacanvas to jaro oxides of nitrogen, texdos.
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  They are fundamentally a direct result of the system just not
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  having enough time, not enough time to reach thermodynamic
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  equilibrium. So before we can figure out why
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  our real fast systems aren't getting there, we really need to
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  define that ideal state mathematically.
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  OK, so let's build up some intuition first, like before
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  diving straight into the heavy math, the concept of
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  equilibrium. It's often easier to grasp with
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  simple mechanical systems. You know, the classic marble
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  analogies. Yeah, those are great.
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  They really set the stage for the the more rigorous
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  mathematical criteria we'll need later, especially when we talk
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  about entropy and energy. Right.
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  So starting with the first one, stable equilibrium, just picture
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  a marble sitting right at the bottom of a smooth bull.
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  If you give it a little nudge, what happens?
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  It rolls right back down to the bottom.
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  It wants to be there. And mathematically, this
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  stability translates to two conditions for an isolated
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  system, the total variation of entropy, which we write as text
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  dao, well, that has to be negative, so text dollar and at
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  the same time the corresponding variation of energy text has to
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  be positive. Essentially, if you disturb it,
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  the system's energy goes up slightly and that pushes it back
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  towards the stable lower energy state.
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  Makes sense. Then you've got neutral
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  equilibrium. Think of the marble, but this
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  time it's in a perfectly flat horizontal through.
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  The source material tells us it's an equilibrium anywhere
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  along that through axis. Which means the criteria are 0
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  for both variations in that direction.
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  So text dollars me, but only along the through axis of
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  course. OK, Next up, 3 unstable
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  equilibrium. This is like trying to balance
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  that marble right on the very top point of a smooth hill or
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  Dome. Right, precarious.
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  And the math shows the opposite signs compared to stable here.
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  Tex Dillers and any tiny movement.
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  Any nudge at all and it just rolls further away from that
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  supposed equilibrium point. It doesn't return.
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  And finally, there's four meta stable equilibrium.
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  This one's a bit more subtle. Imagine the marble is sitting in
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  a little dip, but that dip is actually higher up than another
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  deeper dip nearby, like the higher of two troughs.
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  Tiny disturbances. It just rolls back to the bottom
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  of its little dip. Exactly infinitesimal variations
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  cause it to return, but if you give it a bigger push.
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  A larger variation. You can get knocked over the
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  edge and fall down into the lower, more fundamentally stable
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  through. OK, these analogies are helpful.
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  Now you mentioned the notation text Eagle.
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  Why the a? Why not just?
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  Good question. It's a crucial detail actually.
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  When we talk about equilibrium conditions, often the 1st order
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  change, the differential text might be 0.
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  That tells you you're at an equilibrium point, maybe stable,
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  maybe unstable, maybe neutral. But for stability you need to
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  look beyond the first term. Text represents the total
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  variation, which includes higher order terms from like a Taylor
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  series expansion. So mathematically, text plus
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  FRAX 2 plus whole thing, text 2 plus not zero usually because
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  those second derivative terms and so on are not 0.
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  That's why for a truly stable state, the requirement is that
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  the total variation of entropy must be negative.
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  It accounts for the curvature if you like, not just the slope.
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  OK, I get it. So text dollar finds the peaks
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  and valleys, but text dollar confirms it's definitely a
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  valley, a stable point. Precisely.
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  Hasavilis and Keenan summarize the derived criteria nicely.
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  Stable equilibrium has EDS, neutral has EDSE and unstable
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  has EDS and unstable has EDS but the eye straights.
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  EDS, right? And that link you made earlier,
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  the engineer's intuition that energy rises upon disturbance in
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  a stable mechanical system. That's mirrored by this negative
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  variation of entropy in thermodynamics, yes.
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  That's the conceptual bridge. OK, let's use that bridge.
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  Let's transition now to a purely thermodynamic system, the
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  classic example, heat transfer between two blocks.
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  OK. Imagine an isolated system, just
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  two metal blocks, a initially different temperatures text T91
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  and text T2. Let's assume text T is hotter
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  than text T2. They're connected by some
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  conductor, so heat and amount of energy delta of kiora is
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  transferred from the hot block to the cold block.
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  Now, if we imagine this transfer happening reversibly, which is
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  sort of an idealized process. Infinitely slowly.
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  Then the entropy changes for each block are defined very
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  precisely by the equations in the source.
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  For block one losing heat text 1 frac delta T1 that's equation
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  1.1, and for block 2 gaining heat plus frac delta text T1.
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  But of course, in reality, because text T1 is actually
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  greater than text T2, the heat transfer is spontaneous.
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  It's irreversible. This means the actual entropy
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  changes are greater than the reversible case, so the
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  inequalities apply. Text 1 frac alta Text T1 which
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  is 1.1 A and text 2 plus frac delta text T2 which is 1.2 a.
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  OK, so for the real irreversible process, what's the total
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  entropy change for the whole isolated system?
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  It's just the sum text 1 plus text 2 rate.
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  Exactly, and when you add the reversible components you get
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  the change left to Frag Alta Text 2.
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  We can rewrite that slightly as texted to TM.
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  And since we started with text 2 and text 2 and not a 2, is
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  positive heat transfer that whole term?
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  Must be positive. The total entropy, the isolated
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  system increases during this spontaneous process.
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  That's the second law in action. It confirms the system naturally
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  moves towards increasing entropy.
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  Now how do we prove that the equilibrium state itself
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  corresponds to the maximum possible entropy we?
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  Can use a sort of proof by contradiction.
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  Imagine the blocks are already in equilibrium, so $2.00 of t =
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  t two equals is T. Now suppose hypothetically a
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  tiny amount of heat delta Q spontaneously transferred
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  reversibly due to some infinitesimal fluctuation delta
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  T, say from T plus delta T to T delta OK.
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  Following the math in the source, the entropy changed Tait
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  would be the heat out over the slightly higher temp plus the
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  heat in over the slightly lower temp.
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  So text delta T plus DD T. Right, and if you combine those
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  fractions an approximate for small delta T ignoring the tiny
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  delta T2 term in the denominator.
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  You get equation 1.4 from the text.
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  Take two plus delta T plus delta T plus delta T facts.
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  Take Delta TT. Hold on, there seems to be a
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  signed discrepancy in the source text.
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  Equation 1.4 derivation step. Let's reevaluate.
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  Frac delta T plus frac delta T plus frac delta T.
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  Common denominator is D plus delta TTT 2D T 2 numerator
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  becomes delta Q delta T plus delta Q plus delta Q plus delta
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  T so frac and TQ delta T delta T It's approximate for small delta
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  T. This is approx frac to Q delta
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  T2. OK, the source materials
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  intermediate step had a sign flip, but the ultimate
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  conclusion is what matters. If we assume heat delta Q flows
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  from hotter delta T plus delta T2, then delta Q and delta T are
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  both positive I. Mean text would be positive
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  frack to T2 delta T2 delta T2, which supports the idea that any
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  spontaneous fluctuation away from uniform temperature
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  increases entropy, implying the uniform state was not the
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  maximum. Wait, the outline says the
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  result proves maximum entropy is obtained when T1T2 because
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  entropy would have decreased. Let's reconsider the sources
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  intent with equation 1.4. Even with the apparent sign
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  issue in the intermediate step, the procs Rack 2DD takes.
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  You OK, let's follow the source text.
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  Final form Text rack 2D T 2. If E delta Q spontaneously flows
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  from the slightly hotter part T plus delta T to the slightly
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  cooler part, then delta Q's is positive heat, leaving the plus
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  DT part entering the HDT part and delta T is positive.
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  In this scenario, the sources equation yields a negative text.
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  So a spontaneous fluctuation equation away from equilibrium
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  uniform T would lead to a decrease in entropy according to
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  this equation. That doesn't seem right based on
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  the second law. Agreed, the derivation seems
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  problematic as transcribed or interpreted.
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  However, the conclusion drawn from it in the broader
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  thermodynamic context is sound. The state of uniform temperature
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  T dollar T22 is the state of maximum entropy for this
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  isolated system. Any spontaneous process drives
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  the system towards the state, increasing total entropy until
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  the maximum is reached at equilibrium.
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  Right, let's perhaps rely on the established principle rather
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  than getting bogged down in that specific derivations potential
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  issue. The take away is clear.
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  Equilibrium in an isolated system means maximum entropy.
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  Yes, and this leads to the general criterion stated by
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  Keenan back in 1963. For stability of any system, the
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  variation of entropy shall be negative.
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  Which is written mathematically using that total variation
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  notation again as text total later.
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  That's equation 1.5. And this really gives us the
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  definitive mathematical statements for equilibrium and
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  stability in an isolated system based on entropy.
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  That means spontaneous change will occur.
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  You're not stable. May be unstable.
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  Moving away from equilibrium 2. If ESEE, it means you're at an
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  equilibrium point. Could be stable, neutral or
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  unstable. You're at a flat spot, peak or
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  valley. And if etex EEE, that's the
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  rigorous criterion confirming it's a stable equilibrium.
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  The bottom of the valley. OK, so the entropy criterion is
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  powerful. It's universal for isolated
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  systems, but it's a bit abstract, isn't it?
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  Like engineers usually deal with systems that aren't perfectly
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  isolated. They're often held at constant
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  temperature or constant pressure or constant volume.
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  Exactly. And that's why we developed
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  derived properties, things like Helmholtz and Gibbs energies.
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  They essentially translate that abstract entropy criterion into
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  something more tangible, often related to work potential under
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  those more practical constraints.
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  Right, making it more directly useful for design calculations.
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  Let's start with Helmholtz energy, which we usually denote
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  is $5. OK, what's what's the setup for
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  deriving that? Imagine you have your system,
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  system A. It can exchange heat delta Q
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  dollars and do work DD Q. The first law for just system A
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  is pretty standard delta text to Dole plus delta text you qua.
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  That's equation 1.6 E Here is the total energy of the system.
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  Now imagine this system interacts with a large heat
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  reservoir at a constant temperature, $2.00 dollars.
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  We want to find the maximum possible network we can get out
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  of this combined setup, maybe using a reversible engine
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  between A and the reservoir through a derivation involving
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  the entropy changes of both the system and the reservoir, and
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  applying the second law, we find something really neat.
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  The maximum network obtainable delta W textnet turns out to be
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  equal to the decrease in a specific state function of the
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  system. A and that state function is.
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  It's ET dollars SOA, where E is the system's total energy, S is
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  entropy, and T dollars is the reservoir temperature.
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  So the equation is Delta textile to Lamont stellar OK.
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  So the maximum work you can extract is equal to the drop in
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  this specific combination of properties.
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  Precisely, and that combination ETSC evaluated the system's
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  temperature T, often assumed equal to T dollars for processes
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  occurring at the reservoir. Temperature is what we define as
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  the Helmholtz free energy F. For typical thermodynamic
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  systems where the total energy E is primarily internal energy U,
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  the definition becomes V equals UTSC.
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  That's equation 111. So F reresents the maximum total
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  work you can get from the system during a process at constant
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  temperature T. Yes.
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  And that TT dollar term, you can think of it as the energy that's
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  unavailable for work because it's associated with the
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  system's entropy, the energy that must be sort of discarded
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  as heat to the surroundings to satisfy the second law.
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  OK, that makes sense. Maximum total work.
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  But often, especially in like chemical engineering or
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  processes open to the atmosphere, we care more about
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  the useful work, right? The work that's left after
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  accounting for pushing the atmosphere out of the way.
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  Exactly. If the system changes volume by
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  texte fee against a constant external pressure to go well,
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  like atmospheric pressure, it does work equal to $2.00 text to
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  just on the surround surroundings.
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  That's not usually considered useful.
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  So we need to subtract that from the network.
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  The useful work Delta Wu would be Delta Wu, Delta W, she's you.
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  That's equation 1.12. Right, and now substitute our
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  expression for delta W or text net from before.
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  OK, so delta text Udu text text uzu.
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  How does it simplify? Well, if the process happens at
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  constant pressure P dollars, we can bring P dollars inside the
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  differential as D kdel. So delta text you do.
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  Combining the terms inside the differential gives delta text to
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  you E plus P0VT0 das. And E plus PVD is just enthalpy
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  H, assuming E is mainly U. Exactly.
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  So assuming you refrox you do in the process occurs at T prox T&P
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  prox PU year, the maximum useful work becomes delta text juju we
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  text DVDO. And there it is.
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  That TTC must be the Gibbs free energy G.
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  That's the one defined as DG equals HTS equation 1.14.
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  So G represents the maximum useful work obtainable from a
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  system during a process at constant temperature and
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  pressure. Correct.
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  And these functions UHF and G are incredibly powerful because
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  they lead to a set of fundamental differential
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  relations that connect all the key properties, especially when
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  you include composition changes in multi component systems.
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  The Gibbs equations, yeah, those are central.
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  They are. They define how UHF and G change
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  infinitesimally based on changes in TSPV and the amount of each
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  chemical species another dollars.
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  Let's transcribe them from the source, including the chemical
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  potential term knee text FPP plus some me text P plus some me
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  text T, and the really key one for many applications, text T +
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  V text T That's equation 1.15. Wow.
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  OK, that last one, the text equation, you can immediately
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  see why it's so useful. If you're holding temperature
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  constant text, D dollars. If you're holding pressure
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  constant text dealers. Right.
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  Those first two terms just vanish under those common
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  laboratory or industrial conditions.
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  Leaving just text MUI text beta for a process at constant P&P
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  that simplifies things enormously.
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  It absolutely does, and it leads directly to the practical
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  criteria for equilibrium using these derived properties.
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  So let's summarize those for Helholtz energy F, which is
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  useful when. When volume and temperature are
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  held constant, like reactions in a rigid thermostatic container.
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  In that case, the system will spontaneously change in a way
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  that decreases F until it reaches a minimum value.
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  That minimum F signifies equilibrium.
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  And the criterion for stability at constant V&T is that any
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  small perturbation must increase F.
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  So mathematically, text ination if for stability against small
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  changes away from equilibrium, though the source notes text DAT
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  dollar Z would be the more rigorous total variation
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  criterion. Got it.
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  Minimum F at equilibrium for constant Vt.
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  Now for the more common case, Gibbs energy G.
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  This applies when. When pressure and temperature
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  are held constant, think of reactions in an open beaker or
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  processes and flow systems at steady conditions.
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  Right. And the criteria here are
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  directly analogous to the entropy criteria, but flipped
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  because G tends to a minimum, not a maximum.
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  Exactly. The source lays it out clearly.
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  One if text AG the change is spontaneous, the reaction or
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  process will proceed as written. Moving towards equilibrium.
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  If text EG the system is at equilibrium, no net tendency to
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  change. The bottom of the Gibbs Energy
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  Valley. And for stability the total
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  variation must be positive if tax dollars hours at any
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  fluctuation away from equilibrium must increase G.
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  OK, so minimum G signifies stable equilibrium at constant
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  PNT. Let's connect this back to the
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  real world. Where do we actually encounter
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  these different types of equilibrium?
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  Well, stable equilibrium is what we aim for in what's often
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  called thermostatics. It's in the most common state we
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  analyze. Think about a simple mixture of
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  liquid water and water vapor coexisting in a closed container
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  at a fixed temperature and the corresponding saturation
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  pressure that's stable. Or, you know, basically any
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  point defined on a standard text heat crevy diagram for a pure
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  substance represents a state of stable equilibrium.
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  Right. What about metastable
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  equilibrium? Where does that pop up?
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  All over the place, especially in systems that rely on an
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  initiation step. We mentioned combustion and
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  spark ignition engines earlier. The fuel air mixture before the
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  spark is metastable, it's reasonably stable, won't just
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  ignite on its own, but a small perturbation, the spark energy
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  pushes it over the activation barrier into a much more stable
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  state, the combustion products. Like the marble and the higher
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  through needing a nudge. Exactly.
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  Another classic example is the Wilson cloud chamber.
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  You have water vapour that's super saturated, cooled below
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  its normal dew point without condensing, it's Med has stable.
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  It stays as Vader until a charged particle passes through,
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  creating ions that act as nucleation sites, the
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  perturbation causing condensation trails.
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  Fascinating and unstable equilibrium.
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  Is that something we can actually observe?
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  Not really. Sustained, no.
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  Yeah, it's inherently transitory.
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  It represents states the system passes through on its way to a
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  stable or metastable state. Think about gases mixing.
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  If you have two different gases separated by a partition and you
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  suddenly remove the partition, the initial state of separated
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  gases is unstable with respect to the mixed state.
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  The driving force for mixing is a difference in chemical
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  potential, pushing the system towards the more stable, higher
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  entropy mixed state. Or back to combustion, the gases
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  in the cylinder during the very rapid combustion process itself.
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  They haven't fully reacted yet. Precisely, they are in highly
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  unstable intermediate states, rapidly moving towards the more
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  stable equilibrium products like Texkeo 2 texto.
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  But as we said, they might not get all the way there if the
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  process is too fast, leaving things like Dex TO day.
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  Which brings us back to that Gibbs energy minimum at constant
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  PNT. Why is that so critical,
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  especially for mixtures? Engineers sometimes struggle
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  with visualizing changes at constant PNT, causing a property
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  like G to change. It's because the composition can
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  change even if PNT are fixed. The system isn't necessarily
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  stuck. Take text QO 2.
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  At high temperature and atmospheric pressure, it's not
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  perfectly stable. It could breakdown dissociate
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  into text TO and text O22. That reaction involves changes
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  in the amounts the text QO species according to the text
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  equation at constant P&T. If the current mixture
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  composition doesn't correspond to the minimum possible value of
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  G for that P&T, then text U will be negative for the dissociation
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  reaction and it will proceed spontaneously until that minimum
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  G is reached. The chemical potentials, Dr. The
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  Change. So minimum G dictates the
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  equilibrium composition of a mixture at a given P&T exactly.
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  It's fundamental for chemical equilibrium calculations.
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  OK, let's try and synthesize this.
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  This deep dive has really laid out the bedrock criteria for
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  stability in thermodynamics. Yeah, we started with the
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  universal tendency of isolated systems towards maximum entropy
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  mathematically captured by 9 texted ES or simply detects DLA
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  driving spontaneous change towards equilibrium.
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  And then we saw how derived functions Helmholtz energy F and
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  Gibbs energy G translate this for more practical scenarios.
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  F defines the maximum total work available at constant V&T and
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  its minimum significant equilibrium under those
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  conditions. While G defines the maximum
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  useful work at constant TNT and it's minimum signifies
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  equilibrium there, which is crucial for chemical reactions
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  and phase changes. The minimum energy, maximum
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  entropy principles are the guiding forces.
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  Now our source material dangled a rather interesting thought at
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  the end. It hinted that irreversible
• 20:12
  thermodynamics might be key for future energy tech.
• 20:15
  Yeah, things like efficient solar energy conversion or fuel
• 20:18
  cells. These systems often operate in a
• 20:20
  steady state, not true stable equilibrium.
• 20:23
  They have continuous flows of energy and matter.
• 20:26
  Understanding the thermodynamics of these non equilibrium states
• 20:30
  might unlock new efficiencies. And it leaves us, the design
• 20:34
  junkies listening with a really provocative question, doesn't
• 20:36
  it? I think so.
• 20:38
  If we know that pollutants like text no UPS knows form because
• 20:41
  combustion is too fast to reach the minimum G equilibrium state,
• 20:45
  then maybe the focus shouldn't just be on the equilibrium
• 20:48
  endpoint itself. Maybe the real engineering
• 20:50
  challenge is controlling the rate of approach to equilibrium,
• 20:53
  understanding and manipulating the kinetics, how fast the
• 20:56
  system travels along the path from its initial unstable or
• 21:00
  attestable state towards the stable one.
• 21:02
  If we could control that rate, manage that distance from
• 21:05
  equilibrium effectively, could that be the key to actually
• 21:08
  achieving year 0 emissions? Controlling the journey, not
• 21:11
  just the destination. That definitely sounds like a
• 21:14
  topic for another deep dive. Non equilibrium thermodynamics
• 21:17
  in kinetics. Absolutely.
• 21:19
  It's where a lot of cutting edge research is happening.
• 21:21
  Until the next equation design junkies, we are mechanical
• 21:24
  engineering made simple.