Thermodynamics Take II: The Second Law, Gibbs-Helmholtz Equation, State Functions

This will be the final post about thermodynamics, but related posts (namely on some biophysics-type topics) will be posted down the line.  The equations presented are summarized at the bottom of this post.

How do biological systems follow this second law?  After all, we’re all (rather) organized beings, and there had to be a decrease in entropy when our DNA, lipids, proteins, and all the other biomolecules organized in our bodies.  However, biological systems follow the law because they are open systems and take in (exchange) energy from the environment.  The entropy of the surroundings increases even though the entropy of the system (such as the human body) decreases. 

As mentioned, entropy is a measure of disorder, in a way.  Entropy can be calculated as

S = k_B ln W

where k_B = 1.38x10^-23J/K (Boltzman’s constant) and W is the number of ways to arrange a state.  If you wanted to calculate this, you could, but we are more concerned with changes in entropy than the actual entropy inherent in a molecule.  For example, in the case of glucose (C₆H₁₂O₆) and six oxygen molecules being converted to six molecules of CO₂ and H₂O, the entropy will increase because there are 12 molecules of carbon dioxide and water, but there are only 7 of glucose and oxygen. 

Relationship of Free Energy, Entropy, and Enthalpy

All of the above thermodynamic properties are related in what is called the Gibbs-Helmholtz equation, which states:

ΔG = ΔH – T ΔS

where T is the temperature in Kelvin (always a positive value).  Considering this equation further, one can see that ΔG is negative (a process is spontaneous) if ΔH is negative and ΔS is positive.  On the other hand, if ΔH is positive and ΔS is negative, ΔG is positive and the process is not spontaneous (it would require energy input for it to occur).  In fact, if ΔG is positive, the reverse process is spontaneous (conversion of products to reactants).  If ΔH and ΔS have the same sign, ΔG could be either positive or negative, depending on the magnitude of the values. 

Given the Gibbs-Helmholtz equation, one can quickly calculate the transition temperature at which point a reaction (or process) changes from spontaneous to non-spontaneous as:

T = ΔH / ΔS

which occurs when ΔG = 0. 

Reactions that are considered entropy driven are those that have a positive ΔS and ΔH values and a negative ΔG value, indicating that the reaction is spontaneous and that the change in entropy is the major factor contributing to the spontaneity.  In contrast, an enthalpy-driven reaction is one in which ΔH is negative and ΔS is positive, meaning that ΔG is negative.  In this case, the negative free energy value is due solely to the negative value of the change in enthalpy.

All of the above terms (enthalpy, entropy, free energy) are considered state functions, meaning that the values of enthalpy, entropy, and free energy depend on the system’s current state, not the path to get to that state. Due to this convenient rule, we can calculate the free energy of formation (ΔG_f^o) for various compounds by adding and subtracting free energies of the component molecules at the biochemical standard state (1 atm, 25^oC, pH 7.0). 

One important result of free energy being a state function is that free energy changes are additive: chemical reactions can be “added” (add reactants to reactants, products to products) and the total free energy change is the sum of the component reactions.  

Summary of equations:

ΔG = ΣG_products - ΣG_reactants

ΔH = ΣH_products - ΣH_reactants

ΔS = ΣS_products - ΣS_reactants

ΔS_universe = ΔS_system + ΔS_surroundings > 0

S = k_B ln W

Gibbs Helmholtz Equation: ΔG = ΔH – T ΔS

Transition Temperature: T = ΔH / ΔS

Thermodynamics Take I: The Basics, Free Energy, Entropy, Enthalpy

Today’s post will be a slight departure from cell biology and genetics and will focus, instead, of some of the basics of biochemistry.  It is true that I disliked biochemistry, and that’s putting it lightly, but it’s still something that is important (how important is another question) to understand.  I took a few days off from updating but I return with more review fun.  This set of posts will include a number of equations that I will summarize after all of the thermodynamics notes have been posted.  Also, the illustrations for these posts will be simplistic, but if you have some better ideas of how to illustrate thermodynamics, I'd like to know because I'm coming up with nothing...

Thermodynamics is the study of the relationships between energy and chemical processes.  This energy can come in two distinct forms, namely potential and kinetic energy.  Most of use probably learned in high school physics class that potential energy is stored energy, as in a ball that you hold above the ground  has the potential to fall to the ground and therefore has stored / potential energy.  Kinetic energy is the energy of motion, which you probably learned as the energy of a moving ball during that same physics lesson.  In terms of biology, however, potential energy has deeper meaning (it’s more than balls), such as stored energy in chemical bonds (ATP), concentration gradients, and electrical potential via ion gradients.  Kinetic energy in terms of biology can come in the form of heat energy due to atomic motion (just as we learned in physics) or in radiant energy, including electromagnetic radiation (light).

The First Law of Thermodynamics: energy is conserved

Sure, there are about a billion ways of rephrasing the first law of thermodynamics, but, put simply, energy is conserved.  In terms of chemical reactions, there is what is called the free energy (Gibbs free energy), or G.  Gibbs free energy is the work that is available to do work.  In a reaction or process, the change in free energy is calculated as:

ΔG = ΣG_products - ΣG_reactants

When ΔG is negative (ΔG < 0), the reaction or physical process is spontaneous, though that is not to say that it will happen quickly.  A negative ΔG value simply means that energy need not be added to the system for it to react.  A negative ΔG value is considered exergonic, while a positive ΔG value is endergonic.  A positive value for ΔG means that energy must be added to the system for the process.  At equilibrium in a reaction, ΔG is zero, meaning that neither the amount of products or reactants is changing: no energy is consumed or released.

Free energy, G, can be further broken down into enthalpic, H, and entropic, S, components. 

Enthalpy is a measure of the internal energy of a system in kcal/mol (or kJ/mol).  At constant temperature and pressure, enthalpy is equivalent to the heat absorbed or released and

 ΔH = ΣH_products - ΣH_reactants

Endothermic reactions are those that absorb heat (ΔH > 0); exothermic reactions release heat energy (ΔH < 0).  Adding heat will affect the equilibrium of the reaction (whether it favors products or reactants).  Because endothermic reactions require energy for the reaction to occur, raising the temperature favors the reactants forming products.  Conversely, adding heat to an exothermic reaction will favor the formation of reactants from products because, in fact, heat is a product of the reaction.

Entropy is a measure of the randomness of a system.  While many would dispute this definition, for our purposes it works (and this write-up is not about semantics).  Entropy is measured in cal/mol K (J/mol K).  As you would expect,

ΔS = ΣS_products - ΣS_reactants

The second law of thermodynamics states that entropy of a system and its surroundings always increases in a reaction.  The disorder will tend to a maximum and ΔS > 0.  Therefore, if we want a more ordered system (such as in the polymerization of DNA from dNTPs), we have to add energy.  This may seem counterintuitive to some degree, but one must remember that we are considering the entropy of both the system and its surroundings:

ΔS_universe = ΔS_system + ΔS_surroundings > 0