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