- How is Helmholtz free energy calculated?
- Spontaneous processes
- Solved exercises
- Exercise 1
- Solution
- Exercise 2
- Solution to
- Solution b
- References
The Helmholtz free energy is a thermodynamic potential that measures the useful work of a closed system under constant temperature and volume. Helmholtz's free energy is denoted as F and is defined as the difference of internal energy U minus the product of temperature T and entropy S:
F = U - T⋅S
Since it is energy, it is measured in Joules in the International System (SI), although other appropriate units can also be ergs (CGS), calories or electron volts (eV).
Figure 1. Definition of Helmholtz energy. Source: Pixabay.
The negative variation of the Helmholtz energy during a process equates to the maximum work that the system can do in an isochoric process, that is, at constant volume. When the volume is not kept constant, part of this work can be done on the environment.
In this case, we refer to work in which the volume does not change, such as electrical work: dW = Φdq, with Φ as the electric potential and q as the electric charge.
If the temperature is also constant, the Helmholtz energy is minimized when equilibrium is reached. For all this, the Helmholtz energy is particularly useful in constant volume processes. In this case you have:
- For a spontaneous process: ΔF <0
- When the system is in equilibrium: ΔF = 0
- In a non-spontaneous process: ΔF> 0.
How is Helmholtz free energy calculated?
As stated at the beginning, the Helmholtz energy is defined as "the internal energy U of the system, minus the product of the absolute temperature T of the system, and the entropy S of the system":
F = U - T⋅S
It is a function of temperature T and volume V. The steps to visualize this are as follows:
- Starting from the first law of thermodynamics, the internal energy U is related to the entropy S of the system and its volume V for reversible processes through the following differential relationship:
From this it follows that the internal energy U is a function of the variables S and V, therefore:
- Now we take the definition of F and derive:
- Substituting there the differential expression obtained for dU in the first step, it remains:
- Finally it is concluded that F is a function of the temperature T and the volume V and can be expressed as:
Figure 2. Hermann von Helmholtz (1821-1894), German physicist and physician, recognized for his contributions to Electromagnetism and Thermodynamics, among other areas of science. Source: Wikimedia Commons.
Spontaneous processes
The Helmholtz energy can be applied as a general criterion of spontaneity in isolated systems, but first it is convenient to specify some concepts:
- A closed system can exchange energy with the environment, but cannot exchange matter.
- On the other hand, an isolated system does not exchange matter or energy with the environment.
- Finally, an open system exchanges matter and energy with the environment.
Figure 3. Thermodynamic systems. Source: Wikimedia Commons. FJGAR (BIS).
In reversible processes the variation of internal energy is calculated as follows:
Now suppose a constant volume process (isochoric), in which the second term of the previous expression has zero contribution. It should also be remembered that according to Clausius inequality:
dS ≥ dQ / T
Such an inequality applies to an isolated thermodynamic system.
So for a process (reversible or not) in which the volume remains constant, the following is true:
We will have that in an isochoric process at constant temperature it is satisfied that: dF ≤ 0, as indicated at the beginning.
So the Helmholtz energy F is a decreasing quantity in a spontaneous process as long as it is an isolated system. F reaches its minimum and stable value when reversible equilibrium has been reached.
Solved exercises
Exercise 1
Calculate the variation of the Helmholtz free energy F for 2 moles of ideal gas at a temperature of 300K during an isothermal expansion that takes the system from an initial volume of 20 liters to a final volume of 40 liters.
Solution
Starting from the definition of F:
Then a finite variation of F, called ΔF, will be:
As the statement states that the temperature is constant: ΔT = 0. Now, in ideal gases the internal energy only depends on their absolute temperature, but since it is an isothermal process, then ΔU = 0 and ΔF = - T ΔS. For ideal gases, the entropy change of an isothermal process is written as follows:
Applying this expression:
Finally, the change in Helmholtz energy is:
Exercise 2
Inside a cylinder there is a piston that divides it into two sections and on each side of the piston there are n moles of a monatomic ideal gas, as shown in the figure below.
The cylinder walls are good conductors of heat (diathermic) and are in contact with a reservoir of temperature T o.
The initial volumes of each of the cylinder sections are V 1i and V 2i, while their final volumes are V 1f and V 2f after quasi-static displacement. The piston is moved by means of a plunger that hermetically passes through the two cylinder lids.
It asks to find:
a) The change in the internal energy of the gas and the work done by the system and
b) The variation of the Helmholtz energy.
Solution to
Since the piston moves quasi-statically, the external force applied to the piston must balance the force due to the pressure difference in the two sections of the cylinder.
Figure 4. Variation of free energy F in a cylinder with two chambers. Source: F. Zapata.
The work dW done by the external force F ext during an infinitesimal displacement dx is:
Where the relation dV 1 = - dV 2 = a dx has been used, where a is the area of the plunger. On the other hand, the variation of the Helmholtz energy is:
Since the temperature does not change during the process, then dT = 0 and dF = - PdV. Applying this expression to each section of the cylinder we have:
Being F 1 and F 2 the Helmholtz energies in each of the chambers.
The finite work W can be calculated from the finite variation of the Helmholtz energy of each chamber:
Solution b
To find the change in Helmholtz energy, the definition is used: F = U - T S. Since in each chamber there is a monatomic ideal gas at constant temperature T o, the internal energy does not change (ΔU = 0), so that: ΔF = - T or ΔS. Also:
ΔS = nR ln (V f / Vi)
That when substituting finally allows the work done to be:
Where ΔF total is the total variation of the Helmholtz energy.
References
- Chestnuts E. Free energy exercises. Recovered from: lidiaconlaquimica.wordpress.com
- Libretexts. Helmholtz Energy. Recovered from: chem.libretexts.org
- Libretexts. What are Free Energies. Recovered from: chem.libretexts.org
- Wikipedia. Helmholtz energy. Recovered from: es.wikipedia.com
- Wikipedia. Helmholtz free energy. Recovered from: en.wikipedia.com