# Why is energy the ultimate determinant of spontaneity

1. Jul 31, 2008

### biophysics

I understand that a chemical reaction is spontaneous if it has a (-) gibbs free energy = if it is endergonic. My question is:

Why is energy the ultimate determinant of spontaneity

2. Aug 1, 2008

### Mapes

Re: Energy

Actually, the maximization of entropy is the ultimate determinant of spontaneity. This is one of the interpretations of the Second Law. By juggling a few partial derivatives, one can show that the maximization of entropy (i.e., $dS=0$ and $d^2S<0$) is equivalent to the minimization of energy (i.e., $dU=0$ and $d^2U>0$). Or, if you're working at constant temperature and pressure, the minimization of Gibbs free energy. See, for example, Callen's Thermodynamics and an Introduction to Thermostatistics.

3. Aug 1, 2008

### biophysics

Re: Energy

I'm wondering WHY energy or entropy are the most important considerations? Is this a fundamental fact that is taken to be true or is there an even more generalized reason why we always consider energy/entropy

thanks

4. Aug 1, 2008

### Mapes

Re: Energy

I'd say that it's a fundamental fact that entropy maximization is connected to spontaneity. You can come at it from a few different directions. If you define entropy as the conjugate variable to temperature, it's an experimental observation that the entropy of the universe increases during any real process. Alternatively, if you use the statistical mechanics definition of entropy and play with a model system like coins or dice, you will observe that entropy tends to increase when you flip the coins or roll the dice, and that the exceptions become rarer as the system gets larger. Extrapolating that tendency to large systems (>1010 atoms and a far, far larger number of possible microstates) gives the Second Law. Finally, you can simply postulate that entropy is that parameter that is maximized during a spontaneous process and work from there. The results are identical.

5. Aug 1, 2008

### Andy Resnick

Re: Energy

Entropy gets folded into the free energy- DG < 0 for a spontaneous process, DS > 0 for an irreversible process.

Entropy is a fairly fundamental concept, with a lot of interpretations: workless dissipation, distribution of microstates, information/randomness, etc. The (Gibbs) free energy is also a fundamental quantity for isothermal and isobaric systems- conditions that typically hold for biological systems. Saying that these are 'important' considerations simply reflects the fundamental nature of the concept.

To paraphrase in terms of mechanics, "F = ma" doesn't always hold, the conservation of energy does.

6. Aug 1, 2008

### Ygggdrasil

Re: Energy

Lets begin with one of the definitions of ΔG:
$$\Delta G = \Delta H - T\Delta S$$

Recall that:
$$\Delta H = \Delta U + \Delta (PV)$$, $$\Delta U = q + w$$, and $$w = - \int{P dV} + w_{non-pv}$$
(wnon-pv represents non pressure-volume sources of work, for example, electrical work)

Combining these expressions, we have:
$$\Delta G = - \int{PdV} + w_{non-pv} + q + \Delta (PV) - T\Delta S$$

Assuming constant pressure,
$$\Delta G = -P\Delta V + w_{non-pv} + q + P\Delta V -T\Delta S = w_{non-pv} + q -T\Delta S$$

or

$$q = \Delta G -w_{non-pv} +T\Delta S$$

Recall the Clausius inequality:
$$\Delta S \geq \int{\frac{dq}{T}}$$

Assuming constant temperature, this simplifies to:
$$\Delta S \geq q/T \Rightarrow q \leq T\Delta S$$

Therefore,
$$\Delta G - w_{non-pv} + T\Delta S \leq T\Delta S$$

or

$$\Delta G \leq w_{non-pv}$$

Therefore, for any process at constant temperature and pressure, ΔG can only ever stay the same or decrease in the absence of an external source of work. Because ΔG can never increase under these conditions, any process for which ΔG < 0 is irreversible because reversing the process would require ΔG > 0, which is impossible unless an outside source of work is acting on the system.

As you can see, the derivation here uses both the conservation of energy and the Clausius inequality (a consequence of the second law of thermodynamics), so both principles are at play here.

7. Aug 2, 2008

### GCT

Re: Energy

All of the previous explanations encompassed what gibbs is all about , if it doesn't answer your question then you may be seeking a more philosophically oriented answer and my answer to you is that all of these concepts are all about the math . Also if you want to know whether a reaction would happen then the criteria for that is when the gibbs is a non zero value .