S = k ln W

A vignette….

Let’s consider a simpler case first

Thought experiment: Consider a beaker with a partition right in the middle that starts out holding molecules in only one half. What happens when you remove the partition?

• What is DH for this process?

• Zero (at least if molecules don’t interact, “ideal solution”)

So why does it happen?

How about the reverse

process?

[Relevance to drug

distribution]

How about the reverse process?

Thought experiment: Molecules spontaneously moving to only one half of beaker?

• DH is still zero

• This seems unlikely, except maybe if there are very few particles

• Let’s calculate the probabilities quantitatively

Probability of being on one side

1 mol

N = 1p = 1/2

N = 2p = 1/4

Np = (1/2)N

N ~ 6 x 1023

p = (1/2)N = very very small

This fits with our intuition (and the second law) which says that things tend toward disorder, i.e., higher entropy [unless there is an input of energy to counteract …]

Statistical thermodynamics tells us how to compute entropy

• There are multiple ways to define entropy, including the one on Boltzmanns tombstone

In modern notation, this is typically written S = kB ln W

W is the number of configurations available to a system, and is related to the probabilities we just calculated.

Note that S is a state function, and positive values of DS are favorable, not negative like DH. Units on S are J/K (more on this later).

Let’s compute DS for this processThis is not just a thought experiment. This is equivalent to entropy change upon dilution, or mixing two solutions.

State 1 State 2

S1 = kB ln W1 S2 = kB ln W2

N = 1DS = kB ln 2

= 1.38x10-23 J/K * 0.693 ~ 10-23 J/K

We can define the number of “configurations” of the system in different ways, but regardless, W clearly doubles for a single molecule when we double the volume.

To get units of energy, multiply by temperature (300 K),which gives 3x10-21 J. TINY amount of energy.

N = 1DS = kB ln 2

N = 2DS = kB ln 4

We assume that the 2 molecules are uncorrelated, i.e., the configuration of one is unrelated to the other.

(This is again the definition of an “ideal solution”.)

This then implies that the total number of configurations is the product (not sum) of the configurations for each molecule.

N = 1DS = kB ln 2

N = 2DS = kB ln 4

N moleculesDS = kB ln 2N =NkBln 2

N = 1DS = kB ln 2

N = 2DS = kB ln 4

N = NA = 1 moleDS = NAkBln 2 = R ln 2

N = # of moleculesNA = Avogadro’s numberkB = Boltzmann’s constantR = NAkB = gas constant

N = 1DS = kB ln 2

N = 2DS = kB ln 4

N = NA = 1 moleDS = NAkBln 2 = R ln 2

Generalizing to any volume change, Molar entropy:

N = # of moleculesNA = Avogadro’s numberkB = Boltzmann’s constantR = NAkB = gas constant

V = volumeN = # of moleculesn = # of molesR = NAkB = gas constant

We can also think of this in terms of concentrations

State 1 State 2

Convert to concentrations, in molarity units: M = n/V, so V = n/M

There are other flavors of entropy

• We focused here on “translational” entropy, which is equivalent to entropy of dilution, or entropy of mixing

• There is also “rotational” entropy and “vibrational entropy”, very important in drug binding as well

Free Energy• At constant pressure, the correct criterion is

DG < 0where G = H – TS

• If we additionally assume constant T, you get the famous equation

DG = DH – TDS• Key point: note the sign on the entropy term• When DG = 0, this is called “equilibrium”• Molar free energy for a substance also referred to as

“chemical potential”:

An intuitive feel for free energy?

• Enthalpy = energy • Entropy = disorder• Free energy = ??

• Best understood as a statement of the 2nd law of thermodynamics

Free energy of dilution/mixingState 1 State 2

Recall that …

Derivation of DG=-RTln K for a simple AB reaction

BA At equilibrium (constant T, P):

Assume ideal solution, Mº = 1 molar (to

make argument of the logarithm unitless)

Equating these two:

Constant that depends only on nature of

reactants and products

DEFINE KM= (MB/MA)eq

Extension to more general equilibria:

aA + bB « cC + dD

BADC badc

At equilibrium (constant T, P; closed system):

eq

ba

dc

C BA

DCK

][][

][][Alternate Notation

Other concentration units

Derivations are extremely similar.

For molality (mol/kg solvent), the chemical potential can be written as

where the standard state is now defined by mº = 1 molal. The equilibrium constant winds up looking like

A

Bm m

mK

mK-RTG ln 0

Expression for gasses is also very similar, with partial pressures replacing concentrations.