Exponential process

An exponential process is an operation in which a quantity changes exponentially . One distinguishes between

• exponential growth , in which a size grows faster and faster, and
• exponential approximation, in which a quantity approaches a given fixed value. Practically the most important special case of this is exponential decay, in which a variable, monotonically decreasing, approaches zero more and more slowly.

Mostly it's about time changes.

Exponential growth

→ Main article : Exponential growth

If in a growth process of a size the growth rate (i.e. the positive change in size over time) is proportional to the size itself, then there is exponential growth: ${\displaystyle A}$ ${\displaystyle {\tfrac {\mathrm {d} A}{\mathrm {d} t}}}$ ${\displaystyle A}$

${\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}\sim A}$

With the constant of proportionality , the differential equation is obtained from this proportionality relation ${\displaystyle \tau }$

${\displaystyle \tau \cdot {\frac {\mathrm {d} A}{\mathrm {d} t}}=A}$

whose solution is an exponential function :

${\displaystyle A(t)=A_{0}\cdot \mathrm {e} ^{\frac {t}{\tau }}}$

Thus gets the importance of a period in which the size of each of the e grows fold. is the value of the quantity at the beginning (at time ). ${\displaystyle \tau }$ ${\displaystyle A}$ ${\displaystyle A_{0}}$ ${\displaystyle A}$ ${\displaystyle t=0}$

Exponential decay

Exponential decay of a decaying amount of substance of a radioactive nuclide with half-life

If the decrease in a quantity is proportional to the respective value of the quantity itself, one speaks of exponential decay , exponential decrease or exponential decrease .

Examples

Time exponential decay:

• Radioactive decay : every second a fixed percentage of the substance's atomic nuclei decays; the fewer nuclei there are, the more slowly their number decreases
• Discharge of a capacitor through a resistor
• Self induction voltage for voltage changes to a coil
• Amperage when switching off a coil
• Oscillation amplitude of a damped pendulum (with Stokes friction )
• Emptying a water container through a thin hose on the floor: The lower the water level, the lower the water pressure in the hose and the slower the water flows out
• Catalytic degradation of substances through a chemical reaction , see exponential function # chemistry , kinetics (chemistry) # first order reactions .
• Relaxation (NMR) : Reconstruction of the longitudinal or decay of the transverse nuclear magnetization after a disturbance.

Spatially (with the penetration depth) exponential decrease:

• Absorption of some radiation in homogeneous material

Mathematical representation

Since the decrease is a negative change, the differential equation (written here for decrease in time) reads now

${\displaystyle -\tau \cdot {\frac {\mathrm {d} A}{\mathrm {d} t}}=A}$ (it is common to take a positive and write the sign in the equation) ${\displaystyle \tau }$

and whose solution is

${\displaystyle A(t)=A_{0}\cdot \mathrm {e} ^{-{\frac {t}{\tau }}}}$

${\displaystyle \tau }$Thus, the time period in which the size of each of the  (about 37%) drop-fold. It is called time constant , in physics also called lifetime . ${\displaystyle A}$ ${\displaystyle {\tfrac {1}{\mathrm {e} }}}$ ${\displaystyle \tau }$

A more descriptive quantity instead of is the half-life . It indicates the time span within which the size always decreases by half, and can easily be calculated from the time constant: ${\displaystyle \tau }$

${\displaystyle T_{\text{1/2}}=\ln \,(2)\cdot \tau \approx 0{,}6931\cdot \tau }$

Exponential approximation

In many physical processes, a physical quantity balances out between two interconnected bodies / systems.

Exponential approximation to the value 1

Examples:

• The temperature of a metal piece adjusts to the ambient temperature.
• The temperatures of two differently hot, thermally conductively connected metal blocks are equal to each other.
• The voltage of a capacitor to be charged approaches the charging voltage.
• The current intensity when a coil is switched on approaches the current intensity given by Ohm's law .
• The water levels of two differently filled water tanks connected with a thin hose are equal to each other.
• Diffusion : The concentrations of a solute in two interconnected chambers equalize each other.
• The speed of fall of a body in a liquid of finite viscosity approaches its final speed ( Stokes friction ).

Many of these examples have in common that an intensive quantity and an extensive quantity are related to one another:

• Temperature and amount of heat ( thermal energy )
• electrical voltage and electrical charge on the capacitor
• Water pressure and volume (or amount of substance ) in cylindrical containers
• Concentration and amount of substance.

The two quantities are proportional to one another, and a difference in the first quantity causes a flux (or current) of the second quantity to flow between the two systems. This in turn causes a change in the first variable in the systems:

• A temperature difference causes a heat flow and thus temperature changes in both bodies.
• A voltage difference across the capacitor causes an electric current and thus a change in voltage.
• A concentration gradient causes a mass transfer and thus a change in concentration.
• A filling level difference (and thus a pressure difference) causes a material flow and thus a filling level change.

The time change of the intensive size is proportional to the strength of the respective flow, and this is proportional to the difference in size. In such a case, the differential equation applies to a quantity ${\displaystyle A}$

${\displaystyle -\tau {\frac {\mathrm {d} A}{\mathrm {d} t}}=A_{2}-A_{1}}$

This basic state of affairs is the same for the phenomena described above, so knowledge and laws can easily be transferred between them. The laws of diffusion, for example, also apply to heat conduction and electrical charge. (However, electrical phenomena are usually very fast. With liquids / gases without strong friction / damping, the inertia of the moving mass creates additional effects, usually in the form of vibrations and sound waves .)

If one of the two values ​​is constant (outside temperature, charging voltage), the size under consideration will approximate this value. If both values ​​are variable, they will approach each other. In both cases, the values ​​approach a final value that is usually easy to calculate. ${\displaystyle A_{\text{Ende}}}$

One can write as a differential equation

${\displaystyle -\tau {\frac {\mathrm {d} A}{\mathrm {d} t}}=A-A_{\text{Ende}}}$

with the solution

${\displaystyle A(t)=A_{\text{Ende}}+\left(A_{\text{Anfang}}-A_{\text{Ende}}\right)\mathrm {e} ^{-{\frac {t}{\tau }}}}$

It is the value of the beginning (at the time ). ${\displaystyle A_{\text{Anfang}}}$ ${\displaystyle A}$ ${\displaystyle t=0}$

As an approximation to the value 0, the exponential decrease is a special case of the exponential approximation with . ${\displaystyle A_{\text{Ende}}=0}$

The final value A _Ende is never reached, but only increasingly approximated. In practice, the smaller and smaller difference to the final value will eventually become smaller than the measurement inaccuracy. After five times the time constant ( ) the original difference has already dropped to below 1%, after the seven times ( ) to below 1 ‰. ${\displaystyle t=5\tau }$ ${\displaystyle t=7\tau }$

The time constant can be determined in a specific case and depends on variables such as general resistances and capacities. For example, when charging or discharging a capacitor with the capacitance via a resistor with the value : ${\displaystyle \tau }$ ${\displaystyle C}$ ${\displaystyle R}$

${\displaystyle \tau =R\cdot C}$.

See also

• Lifetime (physics)