2.5.2 The Gibbs energy and surface tension
Now that we’ve discussed the Gibbs energy, we can use it to describe
the thermodynamic state of a nanosystem, such as a nanofilm. Since
the surface-area-to-volume ratio in a nanomaterial increases as size
decreases, the surface of the nanomaterial becomes more important in
determining the thermodynamic state of the material. We’ve discussed the
work done in increasing the volume of a gas, but what is the work done in
increasing the area of a surface? Clearly, work must be done to transport a
bulk phase molecule to the surface and thus increase the area of that
surface. This work represents the Gibbs energy change in increasing the
surface area. Strictly speaking, the Gibbs energy of the surface is directly
proportional to the surface area. The proportionality constant is known as
the surface tension of the material in question. We will take a closer look
at surface tension and its determination in Chapters 7 and 8. For now, let’s
define molar Gibbs energy of a surface by Equation 2.53:
G surf = g
A
(2.53)
In the above expression,
A is the molar area (i.e., the number of moles per
unit area on the surface) and g is the surface tension (usually expressed in
N/m).
Later in this chapter, we will need an expression for the surface molar
Gibbs energy in terms of molar volume of a spherical nanoparticle rather
than molar area of a planar surface. In order to derive this expression, we
need to examine how the volume of a spherical particle increases as the
surface area increases. Equations 2.54 and 2.55 give the surface area and
volume of a sphere of radius r, respectively:
A = 4πr
2
(2.54)
V =
4
3
πr
3
(2.55)
We need to know how V increases as a function of time and then compare
the result to how A increases with time. To do this, let’s first take the
derivative of V with respect to time (Equation 2.56):
∂ V
∂ t
=
∂ V
∂ r
Â
∂ r
∂ t
= 4πr
2 ∂ r
∂ t
(2.56)
THE GIBBS ENERGY STATE FUNCTION
47
Now that we’ve discussed the Gibbs energy, we can use it to describe
the thermodynamic state of a nanosystem, such as a nanofilm. Since
the surface-area-to-volume ratio in a nanomaterial increases as size
decreases, the surface of the nanomaterial becomes more important in
determining the thermodynamic state of the material. We’ve discussed the
work done in increasing the volume of a gas, but what is the work done in
increasing the area of a surface? Clearly, work must be done to transport a
bulk phase molecule to the surface and thus increase the area of that
surface. This work represents the Gibbs energy change in increasing the
surface area. Strictly speaking, the Gibbs energy of the surface is directly
proportional to the surface area. The proportionality constant is known as
the surface tension of the material in question. We will take a closer look
at surface tension and its determination in Chapters 7 and 8. For now, let’s
define molar Gibbs energy of a surface by Equation 2.53:
G surf = g
A
(2.53)
In the above expression,
A is the molar area (i.e., the number of moles per
unit area on the surface) and g is the surface tension (usually expressed in
N/m).
Later in this chapter, we will need an expression for the surface molar
Gibbs energy in terms of molar volume of a spherical nanoparticle rather
than molar area of a planar surface. In order to derive this expression, we
need to examine how the volume of a spherical particle increases as the
surface area increases. Equations 2.54 and 2.55 give the surface area and
volume of a sphere of radius r, respectively:
A = 4πr
2
(2.54)
V =
4
3
πr
3
(2.55)
We need to know how V increases as a function of time and then compare
the result to how A increases with time. To do this, let’s first take the
derivative of V with respect to time (Equation 2.56):
∂ V
∂ t
=
∂ V
∂ r
Â
∂ r
∂ t
= 4πr
2 ∂ r
∂ t
(2.56)
THE GIBBS ENERGY STATE FUNCTION
47