• A living cell would represent an "open" thermodynamic system
• A study of the folding or unfolding of a biological polymer in aqueous solution would represent a "closed" system (i.e. no mass is exchanged in a simple phase transition)

"Conservation of Energy"

• Energy may converted into different forms, but energy is neither created nor destroyed
• Energy lost by the system, must be gained by the surroundings and vice versa

Forms of energy

• The internal energy, E, of a system is the sum of all the kinetic and potential energy contained within the system

Changes in the energy of a system

• The change in energy (
DE) of a system refers to the change that occurs when energy flows into, or out of, the system
• Such an energy inflow or outflow will be manifest as a change in the sum of all the kinetic and potential energy contained within the system. If E₁ is the sum of all energy within the initial system, and E₂ is the sum of all energy at some later time, then the change in energy is simply:

DE = E₂ - E₁

• If
DE is positive, it means that energy flowed into the system from the surroundings
• If
DE is negative, it means that energy flowed out of the system into the surroundings

How might such energy be transferred between system and surroundings?

• Generally, either through direct mechanical coupling, or via non-mechanical transfer (i.e. radiation of energy)

DE = E₂ - E₁ = q + w

• "q" is the term that quantifies the non-mechanical transfer of energy
absorbed by the system from the surroundings. PLEASE NOTE THE FRAME OF REFERNCE FOR THE TRANSFER: As defined here, DE is positive when q is positive. A positive value for DE means that the system absorbed energy. Therefore, a positive value for q means that a non-mechanical transfer of energy occurred in the direction from the surroundings to the system. What is the non-mechanical transfer of energy? It is HEAT. A fire warms you up; it is transferring energy to you, but not by some mechanical coupling.
• "w" is the term that quantifies work. It refers to moving an object some distance, against an opposing force. Or, the movement through some distance caused by the application of a force (since force = mass * acceleration, the term for work has implicit in its definition the acceleration of mass over some distance)

w = Force * distance = F * d

• PLEASE NOTE THE FRAME OF REFERENCE
: As defined here, DE is positive when w is positive. A positive value for DE means that the system absorbed energy. Therefore, a positive value for w means that work was done on the system by the surroundings.

The manifestation of "work" in biological systems is often concerned with pressure and volume changes

• Change in volume for a system is defined in the same reference frame as the change in energy:

DV = V₂ - V₁

(where V₂ is final volume, V₁ is initial volume)

• Therefore, a
NEGATIVE value for DV indicates that the surroundings have performed work on the system (i.e. crushed it), and will correspond to a POSITIVE value for w (i.e. work performed on the system).
• Of course, it requires more energy to crush a container if the pressure in the container is high to begin with. Therefore, the quantity of work performed in reducing the volume of the system is directly proportional to the pressure of the system

w = -P * DV

thus

DE = q + (-P * DV)

The change in the internal energy of the system is related to the heat energy transferred to (or from) the system plus any pressure-volume work performed on system (or on surroundings by the system)

• "Work" can take the form of mechanical (moving a molecule or atom), electrical (moving a charged particle or ion), magnetic (deflection of a moving charge in a magnetic field) and chemical (moving electrons and atoms around)
• D
E, q and w have same units: either calorie or joule

DE = q + w. Therefore, if no work is performed during some change in state of a biological system, then the energy transfer is entirely in the form of heat energy (i.e. energy transferred by radiation rather than direct mechanical coupling)

• As defined, no work will be performed under conditions of constant volume (i.e. if
DV = 0 for some change in state, then w = 0). Under this special condition DE for some change in state = q
• Therefore, as a thermodynamic term,
DE is also sometimes referred to as the heat exchanged at constant volume

Tragically, most biological systems have evolved under conditions of constant pressure (i.e. we are under 1 atm pressure, and this is pretty much constant for most "systems" in our body during their thermodynamic changes). Thus, it is not feasible to be able to study and measure biological systems under conditions of constant volume (where we would know that the change in energy is entirely quantitated by the heat transferred between system and surroundings)

• D
E is not a particularly useful thermodynamic term for biological systems

Another thermodynamic term, ENTHALPY (H), has been defined as the following:

H = E + (P * V)

Enthalpy = Internal energy plus pressure-volume contributions to energy

• If pressure is held constant
during some change in the state of the system, then only the volume will change and

DH = DE + (P * DV)

• Since
DE = q - (P * DV)

DH = {q - (P * DV)} + (P * DV)

thus

DH = q

This means that under conditions where pressure is held constant (as with many biological systems) DH is a thermodynamic term that quantitates the heat energy (q) transferred from surroundings to the system.

What about possible volume changes and the work associated with that?

• Since most biological processes occur in the liquid or solid state, and not the gaseous state, volume changes are often minimal (i.e. often little or no PV work is done)
• Thus, enthalpy H and internal energy E are often essentially equal

Standard State Conditions

• Enthalpy (i.e. internal energy when considering biological systems) is a function of the substance in question and how much of that substance you have. Thus, values for H need to be standardized on a concentration basis
• Aqueous solutions of 1mol/liter (i.e. 1 Molar, or 1M) solute are considered the "standard state" for enthalpy calculations. The value of the enthalpy under these conditions is termed "
DH⁰"
• They also require an energetic frame of reference (rather than absolute value). In this case, the enthalpy of elemental forms, or standard forms under conditions of STP are assigned a standard enthalpy of formation (
DH⁰_f = 0) and the molecules of interest will have an enthalpic value for their formation in terms of these reference molecules/states (intrinsic heat energy, q, content)

The van't Hoff enthalpy

If we define conditions where the heat transferred is a measure of the internal energy, then the equilibrium between two thermodynamic states will be changed in response to heat transferred from the surroundings. In other words, the equilibrium will change with the temperature.

Consider an ENDOTHERMIC reaction:

A + heat energy (q) ó B

• A is state 1, B is state 2 and these are ](two different thermodynamic states

K_eq = [B] / [A]

• Note that the equilibrium is affected by temperature. According to Le Chatelier's principle, the input of heat energy into the system will shift the equilibrium to the right.
• From the balanced equation above (which includes the heat energy term q), state 2 must have a higher enthalpy than state 1 (because it requires the input of heat energy to shift the equilibrium from state 1 to state 2)
• Thus, H₂ > H₁, since
DH = H₂ - H₁, DH must be positive for this situation (i.e. DH is positive for an endothermic reaction)
• This is "uphill" energetically (state 2 has a higher energy content that state 1), and considering only the enthalpy term , is expected to be unfavorable (i.e. non-spontaneous)

What is the relationship between K_eq and temperature?

• Higher temperatures are achieved by the input of heat energy, q, into the system
• The input of heat energy, q, should shift the equilibrium to the right (meaning that K_eq should increase with increasing temperatures)

The extent to which K_eq shifts in response to increases in temperature is a measure of the enthalpy change (DH) of the system

• If
DH is a small value, it will not take much of a temperature change to shift the equilibrium (i.e. the input of only a small amount of heat energy, q, is necessary to go from state 1 to state 2)
• The relationship between the change in K_eq and Temperature, and its relationship to
DH for the reaction, is generally given as:

DH⁰ = -R * d(lnK_eq)/d(1/T)

• D
H⁰ is the enthalpy change for the reaction (superscript '0' means under standard conditions of 1M, 1atm)
• D
H⁰ is proportional to how the equilibrium constant for the reaction varies with temperature. The constant of proportionality is the gas constant R
• For the example of an endothermic reaction, we expect that as T increases (note: T is always positive since it is Kelvin), K_eq should increase (i.e. reaction shifts to right). If we were to plot lnK_eq versus 1/T we would expect:

• Note that this is for an endothermic reaction, and the slope of the relationship between d(lnK_eq)/d(1/T) is negative.
• We know that the slope is proportional to
DH⁰ for the reaction (where R is the constant of proportionality) and that, in the above case, DH⁰ should be positive. This results in the following completed description of the relationship (i.e. this is why there is a negative sign in the equation):

DH⁰ = -R * d(ln Keq) / d(1/T)

• Experimental data for the relationship between K_eq and Temp is usually plotted as RlnK_eq versus 1/T (in Kelvin), and
DH⁰ is the value of the NEGATIVE slope

The temperature-dependent denaturation of a biological polymer can be analyzed in this way

• The protein chymotrypsinogen has a Native
ó Denatured equilibrium that is affected by temperature
• D
H⁰ is approximately +500 kJ/mol, i.e. the denatured form (state B) is 500 kJ/mol higher in energy than the folded or native form. Why?
• If the native form has more and better non-covalent interactions (e.g. van der Waals), then energy is required to achieve a state (i.e. the unfolded state) where these are disrupted.
• If the native form has sequestered hydrophobic groups from solvent, then exposing them (as in the denatured state) requires an input of energy

Therefore, a positive DH⁰ value indicates the folded state of the protein is a lower internal energy state than the unfolded state

The internal energy difference (i.e. DH⁰) for protein denaturation varies somewhat with temperature, but is usually always positive (i.e. favors the folded state). However, the enthalpy is not the only contribution to the energetics of protein folding, and proteins do unfold at extremes of temperature.

The Second Law of Thermodynamics

The second law of thermodynamics deals with Entropy (or disorder) and there are several points about entropy that the second law of thermodynamics makes:

• Systems tend to spontaneously proceed from a state of order to a state of disorder
. Since entropy is defined as disorder, this means that systems tend to go from low entropy to high entropy

• Reversible processes (e.g. the freezing and melting of water) also have reversible entropy changes, therefore, there is no net entropy change associated with them. Irreversible processes on the other hand (e.g. a glass breaking on the floor) are associated with a net increases (system + surroundings) in entropy. Since all processes we know of fall into one of these two categories (reversible or irreversible)
the entropy of the universe is always increasing (i.e. because it is made up of processes where the entropy change is either zero or positive)

Here is an example of an irreversible process:

Step 1: I have a flask with two chambers and separated by a valve. One chamber has a gas, the other is evacuated.

Original temp, pressure, but entropy not same as initial value (it is higher)

• All natural processes
proceed to some minimum potential energy level (they go downhill in energy) and they achieve equilibrium at this minimum energy level

 How is entropy quantitated?

• If there are "W" different, but energetically identical, ways to arrange the atoms or molecules in a system, then the entropy, "S" is given as:

S = k * ln(W)

dS_rev = dq/T = dH/T

Cp = dH / dT

The heat capacity at constant pressure = heat energy transferred / change in temperature

• An object with near-infinite heat capacity is using the energy input to essentially increase the entropy of the system. This happens during phase transitions (like with the melting of ice)

Quantitation of entropy in terms of heat capacity

Cp = dH / dT

Rearranging to give enthalpy change in terms of heat capacity and temperature change:

dH = Cp * dT

Substituting into the definition of entropy change in relationship to enthalpy change:

dS = dH/T = (Cp * dT)/T

dS = Cp * (1/T) dT

• If the system is heated from T₁ to T₂,
DS (S₂ - S₁) can be determined by integrating this equation

The equation can be rearranged to describe the heat capacity in terms of the entropy:

T * dS/dT = Cp

Entropy is exactly 0 at 0K

• Therefore, in the integration of the equation for dS, if S₁ = 0 at T₁ = 0 then

• The third law of thermodynamics thus allows us to quantitate entropy of a system, at some temperature T₂, on an absolute level (rather than just quantitating
DS between two temperatures)

If a system gives up heat energy to the surroundings, then the new state of the system is lower in energy. Conversely, if the system absorbed energy from the surroundings, then the new state was higher in energy.

• Systems tend to want to go "downhill" in energetic terms, thus, the release of energy to the surroundings is often observed to be spontaneous (e.g. combustion reactions release heat and are spontaneous)
• However, a few reactions are observed to be spontaneous and also absorb heat (chemical ice packs, for example)
• J. Willard Gibbs (the first person to get a Ph.D. from an American University) realized that you have to understand the contribution not only of enthalpy changes, but also entropy changes, in order to predict spontaneity of a particular reaction. He came up with a new term, called Free Energy (G) that included both terms:

G = H - T*S

DG = G₂ - G₁ = (H₂ - H₁) - T * (S₂ - S₁)

Or

DG = DH - T*DS

(Remember, if P = constant, DH = q)

What does this mean?

• The free energy of a system, G, is equal to the enthalpy of the system, H, plus consideration of the entropy of the system (-T*S)
• While the entropy is some absolute value of disorder, its contribution to energetics is dependent upon the temperature that the system is at. Also, since entropy is defined as disorder, increasing disorder (i.e. large S) reduces the free energy of the system, thus the negative sign on the T*S term. High free energy means high enthalpy (high heat energy content) and low disorder (low entropy)
• For a spontaneous process
DG is negative in sign
• If the system releases energy to the surroundings, then
DH is negative, and is a spontaneous exothermic reaction (spontaneous because the system is going downhill energetically)
• If the system becomes more disordered, then
DS is positive and again DG is negative (due to -T*DS)

Processes that are unfavorable with regard to DH (i.e. endothermic, or absorb heat) can be driven by associated entropic changes that are favorable (i.e. increasing disorder). These are known as "entropy -driven" reactions

Conversely, processes that are favorable enthalpically, but unfavorable entropically, may be spontaneous if the enthalpic term is large enough. These are known as "enthalpy-driven" reactions

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