# Reciprocal proportionality

Reciprocal proportionality , indirect proportionality, inverse proportionality or anti- proportionality exists between two quantities if one is proportional to the reciprocal of the other, or equivalently, the product of the quantities is constant. One variable is then a reciprocally proportional (also anti- proportional ) function of the other variable. The doubling (tripling, halving, ...) of one is associated with halving (thirding, doubling, ...) the other. The function graph is a hyperbola that asymptotically approaches the coordinate axes .

## Reciprocal relationships Function graph of a reciprocal proportional relationship: height and width of rectangles with area = 4 cm
2
${\ displaystyle A}$ The constant product of two quantities and is known from a value pair ( , ). Then one size can be specified as a function of the other: ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle x_ {0}}$ ${\ displaystyle y_ {0}}$ ${\ displaystyle y = {\ frac {A} {x}} = {\ frac {x_ {0} \ cdot y_ {0}} {x}}}$ .

Example: Given is a rectangle, 8 cm wide and 0.5 cm high. We are looking for a rectangle with the same area and a width of 5 cm.
The constant product is 8 cm x 0.5 cm = 4 cm 2 .
The required height is 4 cm 2 / (5 cm) = 0.8 cm.

The diagram opposite shows the two pairs of values ​​as marked points. At the hyperbola you can read off other rectangles of equal area, e.g. B. 1 cm wide, 4 cm high. ${\ displaystyle y = A / x}$ Further reciprocal relationships may be mentioned:

## Reciprocal representation Upper scale linearly divided into lower scale reciprocally divided into divided${\ displaystyle 1 / x}$ ${\ displaystyle x}$ The representation of reciprocal relationships in a Cartesian coordinate system often uses axis labeling , in which the numerical value of a quantity to be represented is not plotted in a linear division , but the reciprocal of its numerical value. Such a representation is particularly helpful when there is a proportionality between the dependent and the reciprocal of the independent variable . This creates a straight curve in a line diagram .

Processes of chemical kinetics of the first order , whose rate constant depends on the temperature according to the Arrhenius equation, should serve as an example

${\ displaystyle k = k_ {0} \ cdot \ mathrm {e} ^ {- {\ frac {E _ {\ mathrm {A}}} {R \ cdot T}}}}$ With

 ${\ displaystyle k}$ Reaction rate constant ${\ displaystyle \ mathrm {e}}$ Euler's number ${\ displaystyle E _ {\ mathrm {A}}}$ Activation energy ${\ displaystyle R}$ universal gas constant ${\ displaystyle T}$ absolute temperature

The equation can be rewritten as

${\ displaystyle \ quad \ ln \ left ({\ frac {k} {k_ {0}}} \ right) = - {\ frac {E _ {\ mathrm {A}}} {R}} \ cdot {\ frac {1} {T}}}$ .

If a process actually proceeds in accordance with the Arrhenius equation as a first order reaction, it can be seen that, in a representation in the above is coated with linear partitions, a straight line is formed, see Arrhenius plot . The activation energy for this straight line results from its rise . ${\ displaystyle \ ln (k / k_ {0})}$ ${\ displaystyle 1 / T}$ ${\ displaystyle (-E _ {\ mathrm {A}} / R)}$ ## Notation

For "a is inversely proportional to b" one writes briefly with one of the two proportionality symbols:

${\ displaystyle a \ sim {\ frac {1} {b}}}$ or   ${\ displaystyle \ displaystyle a \ propto {\ frac {1} {b}}}$ 