Thermodynamic activity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Thermodynamic_activity 

In chemical thermodynamics, activity (symbol a) is a measure of the "effective concentration" of a species in a mixture, in the sense that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution. The term "activity" in this sense was coined by the American chemist Gilbert N. Lewis in 1907.

By convention, activity is treated as a dimensionless quantity, although its value depends on customary choices of standard state for the species. The activity of pure substances in condensed phases (solid or liquids) is normally taken as unity (the number 1). Activity depends on temperature, pressure and composition of the mixture, among other things. For gases, the activity is the effective partial pressure, and is usually referred to as fugacity.

The difference between activity and other measures of concentration arises because the interactions between different types of molecules in non-ideal gases or solutions are different from interactions between the same types of molecules. The activity of an ion is particularly influenced by its surroundings.

Activities should be used to define equilibrium constants but, in practice, concentrations are often used instead. The same is often true of equations for reaction rates. However, there are circumstances where the activity and the concentration are significantly different and, as such, it is not valid to approximate with concentrations where activities are required. Two examples serve to illustrate this point:

• In a solution of potassium hydrogen iodate KH(IO₃)₂ at 0.02 M the activity is 40% lower than the calculated hydrogen ion concentration, resulting in a much higher pH than expected.
• When a 0.1 M hydrochloric acid solution containing methyl green indicator is added to a 5 M solution of magnesium chloride, the color of the indicator changes from green to yellow—indicating increasing acidity—when in fact the acid has been diluted. Although at low ionic strength (< 0.1 M) the activity coefficient approaches unity, this coefficient can actually increase with ionic strength in a high ionic strength regime. For hydrochloric acid solutions, the minimum is around 0.4 M.

Definition

The relative activity of a species i, denoted a_i, is defined as:

${\displaystyle a_{i}=e^{\frac {\mu _{i}-\mu _{i}^{\ominus }}{RT}}}$

where μ_i is the (molar) chemical potential of the species i under the conditions of interest, μ^o
_i is the (molar) chemical potential of that species under some defined set of standard conditions, R is the gas constant, T is the thermodynamic temperature and e is the exponential constant.

Alternatively, this equation can be written as:

${\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln {a_{i}}}$

In general, the activity depends on any factor that alters the chemical potential. Such factors may include: concentration, temperature, pressure, interactions between chemical species, electric fields, etc. Depending on the circumstances, some of these factors, in particular concentration and interactions, may be more important than others.

The activity depends on the choice of standard state such that changing the standard state will also change the activity. This means that activity is a relative term that describes how "active" a compound is compared to when it is under the standard state conditions. In principle, the choice of standard state is arbitrary; however, it is often chosen out of mathematical or experimental convenience. Alternatively, it is also possible to define an "absolute activity", λ, which is written as:

${\displaystyle \lambda _{i}=e^{\frac {\mu _{i}}{RT}}\,}$

Activity coefficient

Main article: Activity coefficient

The activity coefficient γ, which is also a dimensionless quantity, relates the activity to a measured amount fraction x_i (or y_i in the gas phase), molality b_i, mass fraction w_i, amount concentration (molarity) c_i or mass concentration ρ_i:

${\displaystyle a_{ix}=\gamma _{x,i}x_{i},\ a_{ib}=\gamma _{b,i}{\frac {b_{i}}{b^{\ominus }}},\,a_{iw}=\gamma _{w,i}w_{i},\ a_{ic}=\gamma _{c,i}{\frac {c_{i}}{c^{\ominus }}},\,a_{ir}=\gamma _{\rho ,i}{\frac {\rho _{i}}{\rho ^{\ominus }}}\,}$

The division by the standard molality b^o (usually 1 mol/kg) or the standard amount concentration c^o (usually 1 mol/L) is necessary to ensure that both the activity and the activity coefficient are dimensionless, as is conventional.

The activity depends on the chosen standard state and composition scale; for instance, in the dilute limit it approaches the amount fraction, mass fraction, or numerical value of molarity, all of which are different. However, the activity coefficients are similar.

When the activity coefficient is close to 1, the substance shows almost ideal behaviour according to Henry's law (but not necessarily in the sense of an ideal solution). In these cases, the activity can be substituted with the appropriate dimensionless measure of composition x_i, b_i/b^o or c_i/c^o. It is also possible to define an activity coefficient in terms of Raoult's law: the International Union of Pure and Applied Chemistry (IUPAC) recommends the symbol f for this activity coefficient, although this should not be confused with fugacity.

${\displaystyle a_{i}=f_{i}x_{i}\,}$

Standard states

See also: Standard state

Gases

In most laboratory situations, the difference in behaviour between a real gas and an ideal gas is dependent only on the pressure and the temperature, not on the presence of any other gases. At a given temperature, the "effective" pressure of a gas i is given by its fugacity f_i: this may be higher or lower than its mechanical pressure. By historical convention, fugacities have the dimension of pressure, so the dimensionless activity is given by:

${\displaystyle a_{i}={\frac {f_{i}}{p^{\ominus }}}=\varphi _{i}y_{i}{\frac {p}{p^{\ominus }}}}$

where φ_i is the dimensionless fugacity coefficient of the species, y_i is its fraction in the gaseous mixture (y = 1 for a pure gas) and p is the total pressure. The value p^o is the standard pressure: it may be equal to 1 atm (101.325 kPa) or 1 bar (100 kPa) depending on the source of data, and should always be quoted.

Mixtures in general

The most convenient way of expressing the composition of a generic mixture is by using the amount fractions x_i (written y_i in the gas phase) of the different components, where

${\displaystyle x_{i}={\frac {n_{i}}{n}}\,,\qquad n=\sum _{i}n_{i}\,,\qquad \sum _{i}x_{i}=1\,}$

The standard state of each component in the mixture is taken to be the pure substance, i.e. the pure substance has an activity of one. When activity coefficients are used, they are usually defined in terms of Raoult's law,

${\displaystyle a_{i}=f_{i}x_{i}\,}$

where f_i is the Raoult's law activity coefficient: an activity coefficient of one indicates ideal behaviour according to Raoult's law.

Dilute solutions (non-ionic)

A solute in dilute solution usually follows Henry's law rather than Raoult's law, and it is more usual to express the composition of the solution in terms of the amount concentration c (in mol/L) or the molality b (in mol/kg) of the solute rather than in amount fractions. The standard state of a dilute solution is a hypothetical solution of concentration c^o = 1 mol/L (or molality b^o = 1 mol/kg) which shows ideal behaviour (also referred to as "infinite-dilution" behaviour). The standard state, and hence the activity, depends on which measure of composition is used. Molalities are often preferred as the volumes of non-ideal mixtures are not strictly additive and are also temperature-dependent: molalities do not depend on volume, whereas amount concentrations do.

The activity of the solute is given by:

${\displaystyle {\begin{aligned}a_{c,i}&=\gamma _{c,i}\,{\frac {c_{i}}{c^{\ominus }}}\\[6px]a_{b,i}&=\gamma _{b,i}\,{\frac {b_{i}}{b^{\ominus }}}\end{aligned}}}$

Ionic solutions

When the solute undergoes ionic dissociation in solution (for example a salt), the system becomes decidedly non-ideal and we need to take the dissociation process into consideration. One can define activities for the cations and anions separately (a₊ and a_–).

In a liquid solution the activity coefficient of a given ion (e.g. Ca²⁺) isn't measurable because it is experimentally impossible to independently measure the electrochemical potential of an ion in solution. (One cannot add cations without putting in anions at the same time). Therefore, one introduces the notions of

mean ionic activity

aν
± = aν+
+aν−
−

mean ionic molality

bν
± = bν+
+bν−
−

mean ionic activity coefficient

γν
± = γν+
+γν−
−

where ν = ν₊ + ν_– represent the stoichiometric coefficients involved in the ionic dissociation process

Even though γ₊ and γ_– cannot be determined separately, γ_± is a measurable quantity that can also be predicted for sufficiently dilute systems using Debye–Hückel theory. For electrolyte solutions at higher concentrations, Debye–Hückel theory needs to be extended and replaced, e.g., by a Pitzer electrolyte solution model (see external links below for examples). For the activity of a strong ionic solute (complete dissociation) we can write:

a₂ = a^ν
_± = γ^ν
_±m^ν
_±

Measurement

The most direct way of measuring the activity of a volatile species is to measure its equilibrium partial vapor pressure. For water as solvent, the water activity a_w is the equilibrated relative humidity. For non-volatile components, such as sucrose or sodium chloride, this approach will not work since they do not have measurable vapor pressures at most temperatures. However, in such cases it is possible to measure the vapor pressure of the solvent instead. Using the Gibbs–Duhem relation it is possible to translate the change in solvent vapor pressures with concentration into activities for the solute.

The simplest way of determining how the activity of a component depends on pressure is by measurement of densities of solution, knowing that real solutions have deviations from the additivity of (molar) volumes of pure components compared to the (molar) volume of the solution. This involves the use of partial molar volumes, which measure the change in chemical potential with respect to pressure.

Another way to determine the activity of a species is through the manipulation of colligative properties, specifically freezing point depression. Using freezing point depression techniques, it is possible to calculate the activity of a weak acid from the relation,

${\displaystyle b^{\prime }=b(1+a)\,}$

where b′ is the total equilibrium molality of solute determined by any colligative property measurement (in this case ΔT_fus), b is the nominal molality obtained from titration and a is the activity of the species.

There are also electrochemical methods that allow the determination of activity and its coefficient.

The value of the mean ionic activity coefficient γ_± of ions in solution can also be estimated with the Debye–Hückel equation, the Davies equation or the Pitzer equations.

Single ion activity measurability revisited

The prevailing view that single ion activities are unmeasurable, or perhaps even physically meaningless, has its roots in the work of Guggenheim in the late 1920s. However, chemists have never been able to give up the idea of single ion activities. For example, pH is defined as the negative logarithm of the hydrogen ion activity. By implication, if the prevailing view on the physical meaning and measurability of single ion activities is correct it relegates pH to the category of thermodynamically unmeasurable quantities. For this reason the International Union of Pure and Applied Chemistry (IUPAC) states that the activity-based definition of pH is a notional definition only and further states that the establishment of primary pH standards requires the application of the concept of 'primary method of measurement' tied to the Harned cell. Nevertheless, the concept of single ion activities continues to be discussed in the literature, and at least one author purports to define single ion activities in terms of purely thermodynamic quantities. The same author also proposes a method of measuring single ion activity coefficients based on purely thermodynamic processes.

Use

Chemical activities should be used to define chemical potentials, where the chemical potential depends on the temperature T, pressure p and the activity a_i according to the formula:

${\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln {a_{i}}}$

where R is the gas constant and μ^o
_i is the value of μ_i under standard conditions. Note that the choice of concentration scale affects both the activity and the standard state chemical potential, which is especially important when the reference state is the infinite dilution of a solute in a solvent.

Formulae involving activities can be simplified by considering that:

• For a chemical solution:
  • the solvent has an activity of unity (only a valid approximation for rather dilute solutions)
  • At a low concentration, the activity of a solute can be approximated to the ratio of its concentration over the standard concentration:
    $${\displaystyle a_{i}={\frac {c_{i}}{c^{\ominus }}}}$$

    

Therefore, it is approximately equal to its concentration.

• For a mix of gas at low pressure, the activity is equal to the ratio of the partial pressure of the gas over the standard pressure:
  $${\displaystyle a_{i}={\frac {p_{i}}{p^{\ominus }}}}$$

  
  Therefore, it is equal to the partial pressure in atmospheres (or bars), compared to a standard pressure of 1 atmosphere (or 1 bar).
• For a solid body, a uniform, single species solid at one bar has an activity of unity. The same thing holds for a pure liquid.

The latter follows from any definition based on Raoult's law, because if we let the solute concentration x₁ go to zero, the vapor pressure of the solvent p will go to p*. Thus its activity a = p/p* will go to unity. This means that if during a reaction in dilute solution more solvent is generated (the reaction produces water for example) we can typically set its activity to unity.

Solid and liquid activities do not depend very strongly on pressure because their molar volumes are typically small. Graphite at 100 bars has an activity of only 1.01 if we choose p^o = 1 bar as standard state. Only at very high pressures do we need to worry about such changes.

Example values

Example values of activity coefficients of sodium chloride in aqueous solution are given in the table. In an ideal solution, these values would all be unity. The deviations tend to become larger with increasing molality and temperature, but with some exceptions.

Molality (mol/kg)	25 °C	50 °C	100 °C	200 °C	300 °C	350 °C
0.05	0.820	0.814	0.794	0.725	0.592	0.473
0.50	0.680	0.675	0.644	0.619	0.322	0.182
2.00	0.669	0.675	0.641	0.450	0.212	0.074
5.00	0.873	0.886	0.803	0.466	0.167	0.044

 

Del

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Del 

Del operator, represented by the nabla symbol

Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operators depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.

Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for those three operators that makes many equations easier to write and remember. The del symbol (or nabla) can be interpreted as a vector of partial derivative operators; and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product with a scalar, a dot product, and a cross product, respectively, of the "del operator" with the field. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:

• Gradient: ${\displaystyle \operatorname {grad} f=\nabla f}$
• Divergence: ${\displaystyle \operatorname {div} {\vec {v}}=\nabla \cdot {\vec {v}}}$
• Curl: ${\displaystyle \operatorname {curl} {\vec {v}}=\nabla \times {\vec {v}}}$

Definition

In the Cartesian coordinate system Rⁿ with coordinates ${\displaystyle (x_{1},\dots ,x_{n})}$ and standard basis ${\displaystyle \{{\vec {e}}_{1},\dots ,{\vec {e}}_{n}\}}$, del is defined in terms of partial derivative operators as

${\displaystyle \nabla =\sum _{i=1}^{n}{\vec {e}}_{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots ,{\partial \over \partial x_{n}}\right)}$

Where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system R³ with coordinates ${\displaystyle (x,y,z)}$ and standard basis or unit vectors of axes ${\displaystyle \{{\vec {e}}_{x},{\vec {e}}_{y},{\vec {e}}_{z}\}}$, del is written as

${\displaystyle \nabla ={\vec {e}}_{x}{\partial \over \partial x}+{\vec {e}}_{y}{\partial \over \partial y}+{\vec {e}}_{z}{\partial \over \partial z}=\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)}$

Example:

${\displaystyle f(x,y,z)=x+y+z}$

${\displaystyle \nabla f={\vec {e}}_{x}{\partial f \over \partial x}+{\vec {e}}_{y}{\partial f \over \partial y}+{\vec {e}}_{z}{\partial f \over \partial z}=\left(1,1,1\right)}$

Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.

Notational uses

Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.

Gradient

The vector derivative of a scalar field ${\displaystyle f}$ is called the gradient, and it can be represented as:

${\displaystyle \operatorname {grad} f={\partial f \over \partial x}{\vec {e}}_{x}+{\partial f \over \partial y}{\vec {e}}_{y}+{\partial f \over \partial z}{\vec {e}}_{z}=\nabla f}$

It always points in the direction of greatest increase of ${\displaystyle f}$, and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over a plane ${\displaystyle h(x,y)}$, the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.

In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:

${\displaystyle \nabla (fg)=f\nabla g+g\nabla f}$

However, the rules for dot products do not turn out to be simple, as illustrated by:

${\displaystyle \nabla ({\vec {u}}\cdot {\vec {v}})=({\vec {u}}\cdot \nabla ){\vec {v}}+({\vec {v}}\cdot \nabla ){\vec {u}}+{\vec {u}}\times (\nabla \times {\vec {v}})+{\vec {v}}\times (\nabla \times {\vec {u}})}$

Divergence

The divergence of a vector field ${\displaystyle {\vec {v}}(x,y,z)=v_{x}{\vec {e}}_{x}+v_{y}{\vec {e}}_{y}+v_{z}{\vec {e}}_{z}}$ is a scalar field that can be represented as:

${\displaystyle \operatorname {div} {\vec {v}}={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}=\nabla \cdot {\vec {v}}}$

The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.

The power of the del notation is shown by the following product rule:

${\displaystyle \nabla \cdot (f{\vec {v}})=(\nabla f)\cdot {\vec {v}}+f(\nabla \cdot {\vec {v}})}$

The formula for the vector product is slightly less intuitive, because this product is not commutative:

${\displaystyle \nabla \cdot ({\vec {u}}\times {\vec {v}})=(\nabla \times {\vec {u}})\cdot {\vec {v}}-{\vec {u}}\cdot (\nabla \times {\vec {v}})}$

Curl

The curl of a vector field ${\displaystyle {\vec {v}}(x,y,z)=v_{x}{\vec {e}}_{x}+v_{y}{\vec {e}}_{y}+v_{z}{\vec {e}}_{z}}$ is a vector function that can be represented as:

${\displaystyle \operatorname {curl} {\vec {v}}=\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right){\vec {e}}_{x}+\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right){\vec {e}}_{y}+\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right){\vec {e}}_{z}=\nabla \times {\vec {v}}}$

The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centred at that point.

The vector product operation can be visualized as a pseudo-determinant:

${\displaystyle \nabla \times {\vec {v}}=\left|{\begin{matrix}{\vec {e}}_{x}&{\vec {e}}_{y}&{\vec {e}}_{z}\\[2pt]{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\[2pt]v_{x}&v_{y}&v_{z}\end{matrix}}\right|}$

Again the power of the notation is shown by the product rule:

${\displaystyle \nabla \times (f{\vec {v}})=(\nabla f)\times {\vec {v}}+f(\nabla \times {\vec {v}})}$

Unfortunately the rule for the vector product does not turn out to be simple:

${\displaystyle \nabla \times ({\vec {u}}\times {\vec {v}})={\vec {u}}\,(\nabla \cdot {\vec {v}})-{\vec {v}}\,(\nabla \cdot {\vec {u}})+({\vec {v}}\cdot \nabla )\,{\vec {u}}-({\vec {u}}\cdot \nabla )\,{\vec {v}}}$

Directional derivative

The directional derivative of a scalar field ${\displaystyle f(x,y,z)}$ in the direction ${\displaystyle {\vec {a}}(x,y,z)=a_{x}{\vec {e}}_{x}+a_{y}{\vec {e}}_{y}+a_{z}{\vec {e}}_{z}}$ is defined as:

${\displaystyle {\vec {a}}\cdot \operatorname {grad} f=a_{x}{\partial f \over \partial x}+a_{y}{\partial f \over \partial y}+a_{z}{\partial f \over \partial z}={\vec {a}}\cdot (\nabla f)}$

This gives the rate of change of a field ${\displaystyle f}$ in the direction of ${\displaystyle {\vec {a}}}$, scaled by the magnitude of ${\displaystyle {\vec {a}}}$. In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative—the "moving" derivative of the fluid.

Note that ${\displaystyle ({\vec {a}}\cdot \nabla )}$ is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.

Laplacian

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:

${\displaystyle \Delta ={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}=\nabla \cdot \nabla =\nabla ^{2}}$

and the definition for more general coordinate systems is given in vector Laplacian.

The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplace's equation, Poisson's equation, the heat equation, the wave equation, and the Schrödinger equation.

Hessian matrix

While ${\displaystyle \nabla ^{2}}$ usually represents the Laplacian, sometimes ${\displaystyle \nabla ^{2}}$ also represents the Hessian matrix. The former refers to the inner product of ${\displaystyle \nabla }$, while the latter refers to the dyadic product of ${\displaystyle \nabla }$:

${\displaystyle \nabla ^{2}=\nabla \cdot \nabla ^{T}}$.

So whether ${\displaystyle \nabla ^{2}}$ refers to a Laplacian or a Hessian matrix depends on the context.

Tensor derivative

Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field ${\displaystyle {\vec {v}}}$ (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as ${\displaystyle \nabla \otimes {\vec {v}}}$, where ${\displaystyle \otimes }$ represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace of this matrix.

For a small displacement ${\displaystyle \delta {\vec {r}}}$, the change in the vector field is given by:

${\displaystyle \delta {\vec {v}}=(\nabla \otimes {\vec {v}})^{T}\cdot \delta {\vec {r}}}$

Product rules

For vector calculus:

${\displaystyle {\begin{aligned}\nabla (fg)&=f\nabla g+g\nabla f\\\nabla ({\vec {u}}\cdot {\vec {v}})&={\vec {u}}\times (\nabla \times {\vec {v}})+{\vec {v}}\times (\nabla \times {\vec {u}})+({\vec {u}}\cdot \nabla ){\vec {v}}+({\vec {v}}\cdot \nabla ){\vec {u}}\\\nabla \cdot (f{\vec {v}})&=f(\nabla \cdot {\vec {v}})+{\vec {v}}\cdot (\nabla f)\\\nabla \cdot ({\vec {u}}\times {\vec {v}})&={\vec {v}}\cdot (\nabla \times {\vec {u}})-{\vec {u}}\cdot (\nabla \times {\vec {v}})\\\nabla \times (f{\vec {v}})&=(\nabla f)\times {\vec {v}}+f(\nabla \times {\vec {v}})\\\nabla \times ({\vec {u}}\times {\vec {v}})&={\vec {u}}\,(\nabla \cdot {\vec {v}})-{\vec {v}}\,(\nabla \cdot {\vec {u}})+({\vec {v}}\cdot \nabla )\,{\vec {u}}-({\vec {u}}\cdot \nabla )\,{\vec {v}}\end{aligned}}}$

For matrix calculus (for which ${\displaystyle {\vec {u}}\cdot {\vec {v}}}$ can be written ${\displaystyle {\vec {u}}^{\text{T}}{\vec {v}}}$):

${\displaystyle {\begin{aligned}\left(\mathbf {A} \nabla \right)^{\text{T}}{\vec {u}}&=\nabla ^{\text{T}}\left(\mathbf {A} ^{\text{T}}{\vec {u}}\right)-\left(\nabla ^{\text{T}}\mathbf {A} ^{\text{T}}\right){\vec {u}}\end{aligned}}}$

Another relation of interest (see e.g. Euler equations) is the following, where ${\displaystyle {\vec {u}}\otimes {\vec {v}}}$ is the outer product tensor:

${\displaystyle {\begin{aligned}\nabla \cdot ({\vec {u}}\otimes {\vec {v}})=(\nabla \cdot {\vec {u}}){\vec {v}}+({\vec {u}}\cdot \nabla ){\vec {v}}\end{aligned}}}$

Second derivatives

DCG chart: A simple chart depicting all rules pertaining to second derivatives. D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively. Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles (dashed) mean that DD and GG do not exist.

When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v; the use of the scalar Laplacian and vector Laplacian gives two more:

${\displaystyle {\begin{aligned}\operatorname {div} (\operatorname {grad} f)&=\nabla \cdot (\nabla f)\\\operatorname {curl} (\operatorname {grad} f)&=\nabla \times (\nabla f)\\\Delta f&=\nabla ^{2}f\\\operatorname {grad} (\operatorname {div} {\vec {v}})&=\nabla (\nabla \cdot {\vec {v}})\\\operatorname {div} (\operatorname {curl} {\vec {v}})&=\nabla \cdot (\nabla \times {\vec {v}})\\\operatorname {curl} (\operatorname {curl} {\vec {v}})&=\nabla \times (\nabla \times {\vec {v}})\\\Delta {\vec {v}}&=\nabla ^{2}{\vec {v}}\end{aligned}}}$

These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved (${\displaystyle C^{\infty }}$ in most cases), two of them are always zero:

${\displaystyle {\begin{aligned}\operatorname {curl} (\operatorname {grad} f)&=\nabla \times (\nabla f)=0\\\operatorname {div} (\operatorname {curl} {\vec {v}})&=\nabla \cdot (\nabla \times {\vec {v}})=0\end{aligned}}}$

Two of them are always equal:

${\displaystyle \operatorname {div} (\operatorname {grad} f)=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f}$

The 3 remaining vector derivatives are related by the equation:

${\displaystyle \nabla \times \left(\nabla \times {\vec {v}}\right)=\nabla (\nabla \cdot {\vec {v}})-\nabla ^{2}{\vec {v}}}$

And one of them can even be expressed with the tensor product, if the functions are well-behaved:

${\displaystyle \nabla (\nabla \cdot {\vec {v}})=\nabla \cdot ({\vec {v}}\otimes \nabla )}$

Precautions

Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector.

Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is not necessarily reliable, because del does not commute in general.

A counterexample that relies on del's failure to commute:

${\displaystyle {\begin{aligned}({\vec {u}}\cdot {\vec {v}})f&\equiv ({\vec {v}}\cdot {\vec {u}})f\\(\nabla \cdot {\vec {v}})f&=\left({\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}\right)f={\frac {\partial v_{x}}{\partial x}}f+{\frac {\partial v_{y}}{\partial y}}f+{\frac {\partial v_{z}}{\partial z}}f\\({\vec {v}}\cdot \nabla )f&=\left(v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}\right)f=v_{x}{\frac {\partial f}{\partial x}}+v_{y}{\frac {\partial f}{\partial y}}+v_{z}{\frac {\partial f}{\partial z}}\\\Rightarrow (\nabla \cdot {\vec {v}})f&\neq ({\vec {v}}\cdot \nabla )f\\\end{aligned}}}$

A counterexample that relies on del's differential properties:

${\displaystyle {\begin{aligned}(\nabla x)\times (\nabla y)&=\left({\vec {e}}_{x}{\frac {\partial x}{\partial x}}+{\vec {e}}_{y}{\frac {\partial x}{\partial y}}+{\vec {e}}_{z}{\frac {\partial x}{\partial z}}\right)\times \left({\vec {e}}_{x}{\frac {\partial y}{\partial x}}+{\vec {e}}_{y}{\frac {\partial y}{\partial y}}+{\vec {e}}_{z}{\frac {\partial y}{\partial z}}\right)\\&=({\vec {e}}_{x}\cdot 1+{\vec {e}}_{y}\cdot 0+{\vec {e}}_{z}\cdot 0)\times ({\vec {e}}_{x}\cdot 0+{\vec {e}}_{y}\cdot 1+{\vec {e}}_{z}\cdot 0)\\&={\vec {e}}_{x}\times {\vec {e}}_{y}\\&={\vec {e}}_{z}\\({\vec {u}}x)\times ({\vec {u}}y)&=xy({\vec {u}}\times {\vec {u}})\\&=xy{\vec {0}}\\&={\vec {0}}\end{aligned}}}$

Central to these distinctions is the fact that del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function.

For that reason, identities involving del must be derived with care, using both vector identities and differentiation identities such as the product rule.