Thermodynamic Potentials
Thermodynamic Potential
Thermodynamic potentials are used to describe the thermodynamic state of systems, and also describe the equilibrium of the system. The thermodynamic potentials are state functions that depend only on the current state of the system, and is independent of the path taken. The potentials are significant to thermodynamics because they are used to measure the energy of a system in relation to various variables that measure properties of systems. They are referred to as ‘potentials,’ because they, in a way, describe the potential energies that are in the thermodynamic system. Each potential represents a different constraint to the system at hand. The thermodynamic potentials are internal energy (U), enthalpy (H), Gibbs free energy (G), and Helmholtz free energy (A).
Relevant Terms:
 Temperature (T): Measured in Kelvin (K) which is an absolute scale, temperature is the average kinetic energy of atoms; an extensive property. At absolute zero, the particle has minimum motion and is not able to get any colder. If temperature is held constant in a closed system, the free energy (F) of the system decreases until it reaches its minimum value (equilibrium).
 Volume (V): This is measured in the SI unit, liters (L=.001m^{3}) and is an extensive property. The ideal gas law is used to relate volume to other thermodynamic properties (such as pressure and temperature)
 Pressure (P): This is measured in Pascals (10^{5 }Pa= 1 atm). The ratio of the force applied in relation to an area. The force is applied perpendicular to the surface of the object. In relation to thermodynamics, if the pressure is held constant, the enthalpy (H) decreases until equilibrium is reached. SI unit: Pa (Pascals). P = F/A. This is an intensive property:
 Internal energy (U): A measurement in Joules (J). The total energy in a thermodynamic system involving kinetic and potential energy. Kinetic energy includes translational, rotational, and vibrational motion; potential energy is the static energy. The internal energy can be represented by the equation:
where q is heat and w is work. Both work and heat are described as path functions; however, internal energy is a state function.
 Enthalpy (H): A measurement in Joules (J). The total energy in a thermodynamic system (internal energy) as well as the amount of energy necessary to displace the environment to allow for space of the system. A state function and extensive property. When enthalpy is negative, it corresponds to an exothermic reaction (loss of heat) and when enthalpy is positive, it corresponds to an endothermic reaction (gain of heat).
 Gibbs free energy (G): A measurement in Joules (J). Measures the spontaneity of a process under constant pressure and temperature. When ∆G is <0, the process is spontaneous. However, when ∆G is >0, the process is not spontaneous. A state function and extensive property.
 Helmholtz free energy (A): A measurement in Joules (J). Energy needed to create a system minus the energy you can get from the environment. A state function and extensive property.
 Work (w): A measurement in Joules (J). Defined by the amount of force exerted over a distance in the direction of motion. Most often represented as pressurevolume work in thermochemistry: w=P∆V. A path function.
 Entropy (S): A measurement described by Joules per Kelvin (J/K). This is a measure of a system’s thermal energy/T that does not do work. If entropy of a closed system is held constant, the internal energy will decrease until it reaches its minimum value (equilibrium). The entropy of the universe never decreases (). The entropy lost by the system must be less than or equal to the entropy gained by the surroundings given by the second low of thermodynamics: ∆S_{univese}=∆S_{system}+∆S_{suroundings} ≥0. Entropy is a state function.
 Heat (q): A measurement in Joules (J). Defined by the transfer of thermal energy between systems. Is always accompanied by entropy. A path function.
 Heat Capacity (C): A measure of Joules per Kelvin (J/K). The amount of heat required to change a substance’s temperature by a certain amount. An intensive property.
 Specific Heat (c): A measure of Joules per mole times Kelvin (J/molK). Physical property of matter. The heat per mass that is required in order to raise the temperature of a system by 1 degree Celsius.
 State Function: A property of a system that only depends on the current condition. It is independent of that path that brought the system to its current state.
 Path Function: Describes a systems transition between different equilibrium states. Do depend on the path to reach these equilibrium states.
 Extensive Property: Depends on the amount of material in a system.
 Intensive Property: Does not depend on the size of the system and the amount of material in the system
 Microstates: A measure of the number of possible ways a system can be arranged. This number increases with increasing temperature because momentum increases.
First Law of Thermodynamics
The first law of thermodynamics is a statement of the conservation of energy. It states that the energy of a closed system is constant: dU = dQ + dW, where Q is heat and W is work.
For an isolated system, such as the universe, internal energy is 0: dU=0.
Second Law of Thermodynamics
∆S_{univese}=∆S_{system}+∆S_{suroundings} ≥0
The second law of thermodynamics explains that over time, the potential energy of systems will naturally be lower than that of the initial state, regardless of energy entering or leaving the system. In these processes of energy exchange, energy is lost as heat or is deteriorated into unusable energy, and there will be an increase in entropy (S), or disorder and disorganization in the system due to the presence of unusable energy.
This principle also explains that the energy or heat in a system cannot completely be converted into work  100% efficiency (a measure of how well heat is converted to work) is impossible, and this gives rise to an increase in entropy.
When the change of entropy of the universe is positive, this is indicative of a spontaneous process. For example: at over 100◦C the phase change of liquid water to water vapor is spontaneous because the number of microstates increases as the temperature increases, which increases the entropy. When the change of entropy is equal to zero, the system is reversible.
Third Law of Thermodynamics
The third law states that entropy at absolute zero is 0. Because temperature is a measure of the internal energy, if this is zero, then the kinetic energy would also have to be zero. Without kinetic energy, there would be no possible movement of the system.
This law implies that bringing a system to absolute zero is impossible because of the Heisenberg uncertainty principle. The fact that there would be no possible movement implies that there would be a known position and momentum which would violate the Heisenberg uncertainty principle.
Thermodynamic Definitions and Identities
 Internal energy (U): The energy needed to create a system. The total energy in a thermodynamic system involving kinetic and potential energy. Kinetic energy includes translational, rotational, and vibrational motion; potential energy is the static energy. The internal energy can be represented by the equation: ∆U = q + w, where q is heat and w is work. Both work and heat are described as path functions; however, internal energy is a state function.
 Enthalpy (H): The total energy in a thermodynamic system (i.e. internal energy), as well as the amount of energy necessary to displace the environment to allow for space of the system. A state function and extensive property. When enthalpy is negative, it corresponds to an exothermic reaction (loss of heat) and when enthalpy is positive, it corresponds to an endothermic reaction (gain of heat).
 Helmholtz free energy (A): Energy needed to create a system minus the energy you can get from the environment. A state function and extensive property.
 Gibbs free energy (G): Total energy needed to create a system and make room for it minus the energy you can get from the environment. When ∆G is <0, the process is spontaneous. However, when ∆G is >0, the process is not spontaneous. A state function and extensive property.
The thermodynamic potentials of entropy (H), Gibbs energy (G) and Helmholtz free energy (A) are related to internal energy (U), energy from the environment (TS), and pressurevolume work (PV).
They are mathematically defined as such, at constant pressure, temperature, and number of particles:
H = U + PV
A = U  TS
G = H  TS
Gibbs energy can also be expressed as G = U + PV  TS, by substituting the equation of entropy for H.
The thermodynamic potentials can also be expressed more broadly, taking into account changes of pressure, temperature, and the number of molecules.
It can be done so by making use of the thermodynamic identity:
where mew is chemical potential and N is number of particles.
Often times, the number of particles is constant, and the simplified expression is often seen:
We first take a potential, such as entropy:
and take the derivative:
The terms PdV and VdP result from the product rule.
We then substitute the thermodynamic identity for dU; the PdV and +PdV terms cancel:
This leaves us with an expression of the change of entropy with respect to changes in entropy, pressure, and the number of moles.
Similarly, for Helmholtz free energy:
Take the derivative:
Substitute the thermodynamic identity for dU:
We can go a step further and derive expressions of entropy, pressure, and mew in terms of A.
For example, take volume and mole number to be constant. It causes the PdV and udN terms to drop out, for constants have no change, and equate to 0. The equation can then be rearranged to give:
The partial derivative is used because it involves taking the derivative with respect to only one variable, and keeping the other variables as constants, which is exactly what was done here.
The same kind of rearranging of the equation can be done for pressure and chemical potential.
The analysis for Gibbs energy is also given:
These expressions can then be used for computing Helmholtz and Gibbs energy, respectively, at nonfixed temperature and pressure.
The relationships between the different potentials and its variables can be organized in a diagram suggested by Daniel Schroeder (below), which can be used as a tool to memorize these relationships.
Equilibrium
When a system is in thermodynamic equilibrium, all the components are balanced; there is no net change of energy, phase changes, or unbalanced potentials. Objects with different properties which are placed in physical contact interact and their properties change (for example, heat from hot coffee is transferred to the cooler cup). Eventually, these physical changes cause the two interacting objects to reach a state of balance, and the property changes stop (for example, the coffee has transferred its heat to the cup and now both the coffee and the cup have the same temperature. At this point, no more heat is transferred). This is when the system is considered to be at thermodynamic equilibrium.
In order for thermodynamic equilibrium to be reached, the system must meet certain conditions:
 Thermal equilibrium for two systems occurs when when their temperatures are the same. Heat from a hotter object is transferred to a cooler object or its surroundings, like in the coffeecup example mentioned above.
 Mechanical equilibrium for two systems occurs when their pressures are the same. All external forces that act on the system sums up to zero. For example, two gases held in fixed volume containers and separated by an adiabatic piston will have reached mechanical equilibrium when the piston adjusts so that both sides have equal pressure and thus, no net forces.
 Diffusive equilibrium occurs occurs when their chemical potentials are the same and no reactions are occurring in the system. All necessary reactions in the system have occurred or are occurring at an equal rate, so the chemical potentials of both reacting sides are the same.
At equilibrium, a system will have no change in entropy S (which will be at its maximum) at a given point in time; there will be no change in Helmholtz free energy F (which will be at its minimum) when temperature and volume are held constant, and Gibbs free energy G (which will be at its minimum) will show no change at constant temperature and pressure.
We can prove this by using the thermodynamic identity. We have the total entropy defined as the sum of the entropy of the system and the surroundings:
From the thermodynamic identity:
we can rearrange it in terms of dS for the surroundings.
Take volume and the number of particles to be constant, and the expression simplifies further.
By the conservation of energy, the internal energy of the system is negative of that of the internal energy of the surroundings.
We can use the previous two statements for the equation of total entropy:
Take the term 1/T out:
The terms inside the parenthesis is the thermodynamic potential of Helmholtz energy:
This leaves us with an expression in terms of the change of entropy and free energy.
The same type of analysis can be applied for a relationship between entropy and Gibbs free energy by taking constant pressure and temperature.
When the system has no change in entropy S, there will be no change in Helmholtz free energy F when temperature and volume are held constant, and Gibbs free energy G will show no change at constant temperature and pressure.
Ideal Gas Law
The Ideal Gas Law only describes gases that only have perfectly elastic collisions between the atoms or molecules involved in the system. There are also no intermolecular attractive forces present in the. Although the ideal gas law cannot relate to all gas systems, air is similar enough to be represented by the ideal gas law becuase the mean free path is smaller than the distance between the particles of the gas. This allows the collisions between the gas particles to be negligable. The Ideal Gas Law is represented by the following equation:
PV = nRT
In the above equation P represents the pressure (atm), V is volume (Liters, L), n is the amount of moles (mol), R is the rate constant (8.314J/molK, 0.08206Latm/molK). The Ideal Gas Law falls under the assumption that gas particles have no volume, experience no Coulombic forces, and have perfectly elastic collisions.
Within the Ideal Gas Law, there are other laws that are useful. One law is Boyle’s Law which states that pressure is proportional to 1/V where n and T are constant. The following equation displays this relationship:
P_{1}V_{1} =P_{2}V_{2}
Additionally, the picture below provides a visual aid to understand this relationship:
Another law was proposed by Charles GayLussac. He said that the volume and temperature are proportional to another when n and P are held constant. The following equation shows the relationship between the volume and the temperature:
P_{1}T_{1 }= P_{2}T_{2 }
The following picture shows a visual of the Charles and GayLussac’s Law:
A law describing the proportional relationship between volume and moles is Avogadro's Law. Avogadro’s law is only valid providing that the temperature and the pressure are held constant. the following equation exhibits the relationship between the volume and the moles in a system:
n_{1}V_{1} = n_{2}V_{2}
Within Avagadro’s Law, there is molar volume. The molar volume, Vm, is the volume that is occupied per one mole of substance within a system. The following equation shows this relationship:
Another law that relates to Thermodynamics is Dalton’s Law of Partial Pressures. Dalton’s Law of Partial Pressures are the pressures induced by a mixture of ideal gases is the sum of the pressures that each of the gases exerted if they were along in the container at the same pressure of the system.The following equation shows this relationship.
The following picture displays this concept:
Another useful equation for partial pressures is the following:
P_{j} = n_{j}n_{A} + n_{B} + ....P = x_{j}P where x_{j }is the mole fraction.
Real Gases:
The ideal gas has its limits in modeling real gases  real gases follow the ideal gas equation only at low concentrations, low pressures, and high temperatures.
A gas is considered ideal if its compressibility factor, Z = PV/nRT = 1. If Z<1, attractions dominate and if Z>1, repulsions dominate, and is not considered ideal.
The van der Waals equation of state is an empirically derived, correction factor to the ideal gas law to model real gases
a/V^2 represents the would have been pressure in the absence of repulsive interactions, and nb is the effective volume.
Relations of Helmholtz and Gibbs Free Energy to Equilibrium:
How can we use the thermodynamic identities to show that the Helmholtz and Gibbs free energies decrease as the system moves toward equilibrium in a spontaneous process?
Helmholtz free energy:
where A is the Helmholtz free energy, U is internal energy, T is temperature, and S is entropy.
At constant TVN (temperature, volume, amount) the negative of Helmholtz free energy is equivalent to the amount of work that can be extracted from the system.
Gibbs free energy: G = H − TS
Gibbs free energy is described as the free enthalpy to distinguish it from Helmholtz’s free energy. G is gibbs free energy, H is enthalpy, T is temperature, and S is entropy. It is related to constant temperature and pressure of a system. At constant TPN (temperature, pressure, amount) Gibbs Free Energy (the maximum amount of nonexpansion work that can be extracted from a closed system) is minimized.
Equilibrium occurs when the system is balanced and there are no changes in the forces. Therefore, to reach this state of balance, energies must decrease. To decrease energy and reach equilibrium, Helmholtz and Gibbs free energy must be minimized. These two ‘free energies’ want to be minimized because it reduces the amount of work extractable from the system. Since the free energies are directly proportional to temperature, entropy, enthalpy, and internal energy, these thermodynamic functions must also be decreased in order to reach equilibrium.
Applications
Thermodynamic potentials have great relevance to chemistry and biology, and their components play a role in describing spontaneity of reactions, bond formation and dissociation, to name a few. In addition, biological processes depend heavily on the relationships of thermodynamic potentials. Reactions take place everywhere in cells. Cells use energy to push systems out of equilibrium to trigger responses and drives processes forward.
Atoms have their own total energy from their internal structure, potential energy, and kinetic energy. To bond, they must give up some of their total energy; a diatomic molecule is more stable because it total energy is less than the energies of the atoms. Bond formation is spontaneous in the gas phase if and only if the reaction is exothermic, that is, heat is released into the surroundings.
The rate of reactions is dependent on concentration, temperature, and pressure. Higher temperatures equate to a higher average energy of the system, which pushes a reaction forward, and also supplies activation energy necessary to prompt a reaction. Most processes in living systems take place under constant pressure and volume, so it is quite appropriate to use Gibbs free energy to gauge the spontaneity of a reaction.
The following table provides a concrete way to describe the relationship between Gibbs energy and the spontaneity of a reaction.
Enthalpy
Exothermic/
Endothermic

Entropy

Gibb’s Free Energy

Spontaneous/
NonSpontaneous

∆H>0
Endothermic

∆S_{f}>0

If T>∆H/∆S, then ∆G<0

Spontaneous

∆H>0
Endothermic

∆S_{f}>0

If T<∆H/∆S, then ∆G>0

NonSpontaneous

∆H>0
Endothermic

∆S_{f}<0

∆G>0 at all T

NonSpontaneous

∆H<0
Exothermic

∆S_{f}>0

∆G<0 at all T

Spontaneous

∆H<0
Exothermic

∆S_{f}<0

If T>∆H/∆S, then ∆G>0

NonSpontaneous

∆H<0
Exothermic

∆S_{f}<0

If T<∆H/∆S, then ∆G<0

Spontaneous

Problems and Answer Set
 Under what conditions can Gibbs Free energy be minimized?
a.) Constant T, V, and N
b.) Constant moles
c.) Constant T, P, and N
d.) All of the above
2. What is the ideal gas equation and why doesn’t it correctly explain gas behavior?
Answers:
1: c under constant TPN, the maximum amount of nonexpansion work that can be extracted from a closed system, which is Gibb’s free energy, is minimized.
2: PV=nRT, this doesn’t correctly describe gas behavior because it neglects intermolecular attractive forces and assumes purely elastic collisions between molecules. Therefore, it is merely used as a prediction and the Van der Waals equation is more accurate.
Works Cited
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