# density

Physical size
Surname Mass density
Formula symbol ${\ displaystyle \ rho}$
Size and
unit system
unit dimension
SI kg · m -3 M · L −3
relative density (specific gravity),
specific volume

The density ( Rho ), also called mass density , is the quotient of the mass of a body and its volume : ${\ displaystyle \ rho}$ ${\ displaystyle m}$ ${\ displaystyle V}$

${\ displaystyle \ rho = {\ frac {m} {V}}}$.

It is often given in grams per cubic centimeter or in kilograms per cubic meter . In the case of liquid bodies, the unit kilogram per liter is also common. The density is determined by the material of the body and, as an intense variable, is independent of its shape and size.

In general, substances expand with increasing temperature, which causes their density to decrease. An exception are substances with a density anomaly such as B. water .

## Differentiation from other terms

These differences are defined in the DIN 1306 density; Terms, information . The density is a quotient quantity .

## Determination of the density

### Density determination by buoyancy

Forces attacking a submerged body

According to Archimedes' principle , a body completely immersed in a fluid (a liquid or a gas) experiences a buoyancy force that is equal to the weight of the volume of the displaced substance. Two measurements are required to determine the two unknowns, density and volume .

If you immerse any body with its volume completely in two fluids with known densities and , the resulting forces and , which can be measured using a simple balance , result. The required density of the body can be determined as follows: ${\ displaystyle V _ {\ mathrm {K}}}$${\ displaystyle \ rho _ {1}}$${\ displaystyle \ rho _ {2}}$${\ displaystyle F_ {1}}$${\ displaystyle F_ {2}}$${\ displaystyle \ rho _ {\ mathrm {K}}}$

Based on the formulas for the weight of the body and the buoyancy of the body in fluid${\ displaystyle F _ {\ mathrm {G}}}$${\ displaystyle F _ {\ mathrm {A} i}}$${\ displaystyle i}$

${\ displaystyle F _ {\ mathrm {G}} = V _ {\ mathrm {K}} \ cdot \ rho _ {\ mathrm {K}} \ cdot g}$
${\ displaystyle F _ {\ mathrm {A} i} = V _ {\ mathrm {K}} \ cdot \ rho _ {i} \ cdot g}$

With the acceleration of gravity , a balance measures the force of the body immersed in fluid${\ displaystyle g}$${\ displaystyle i}$

${\ displaystyle F_ {i} = F _ {\ mathrm {G}} -F _ {\ mathrm {A} i}.}$

From these two equations for the fluids ( ) one can eliminate the unknown volume and obtain the solution: ${\ displaystyle i = 1,2}$${\ displaystyle V _ {\ mathrm {K}}}$

${\ displaystyle \ rho _ {\ mathrm {K}} = {\ frac {F_ {1} \ cdot \ rho _ {2} -F_ {2} \ cdot \ rho _ {1}} {F_ {1} - F_ {2}}}}$

If one density is much smaller than the other (for example with air and water), the formula simplifies to: ${\ displaystyle \ rho _ {1} \ ll \ rho _ {2}}$

${\ displaystyle \ rho _ {\ mathrm {K}} = {\ frac {F_ {1}} {F_ {1} -F_ {2}}} \ cdot \ rho _ {2}}$

If you only have one liquid, e.g. B. has water with density , the volume of the body can instead be determined by the volume of the water that is displaced when completely immersed, for example by measuring the overflow from a full vessel with a measuring cylinder . From the above equation ${\ displaystyle \ rho _ {1}}$

${\ displaystyle F_ {1} = F _ {\ mathrm {G}} -F _ {\ mathrm {A} 1} = V _ {\ mathrm {K}} \ cdot g \ cdot (\ rho _ {\ mathrm {K} } - \ rho _ {1})}$

is obtained by forming:

${\ displaystyle \ rho _ {\ mathrm {K}} = \ rho _ {1} + {\ frac {F_ {1}} {V _ {\ mathrm {K}} \ cdot g}}}$

Archimedes used this method to determine the density of the crown of a king who doubted that it was really made of pure goldK  = 19320 kg / m 3 ).

The hydrometer (spindle) and Mohr's balance are based on this buoyancy weighing of liquids .

### Other methods

• Pycnometer , determination of the density of solids or liquids by measuring the displaced liquid volumes
• Isotope method , density determination by radiation absorption
• Flexural oscillator , density determination, in particular of liquid flowing through, by measuring oscillations
• Resistograph , determination of the density of wood via strength.
• Floating method , density determination through equilibrium determination with the help of a heavy liquid

A simple estimate of the density can be obtained using the Girolami method .

## Density of solutions

The sum of the mass concentrations of the components of a solution gives the density of the solution by dividing the sum of the masses of the components by the volume of the solution.

${\ displaystyle \ rho = {\ frac {1} {V}} \ sum _ {i} m_ {i} = {\ frac {\ sum _ {i} \ rho _ {i} V_ {i}} {V }}}$

Here are the individual component masses, the individual partial volumes and V the total volume. ${\ displaystyle m_ {i}}$${\ displaystyle V_ {i}}$

## Location-dependent density

The mass in a certain control volume is denoted by. If the mass is continuously distributed, a limit crossing can be carried out, i.e. In other words, the control volume is made smaller and smaller and the mass density can be passed through ${\ displaystyle \ mathrm {d} m}$${\ displaystyle \ mathrm {d} V}$${\ displaystyle \ rho (\ mathbf {x})}$

${\ displaystyle \ mathrm {d} m = \ rho (\ mathbf {x}) \, \ mathrm {d} V}$

define. The function is also known as a density field . ${\ displaystyle \ rho \ colon \ mathbb {R} ^ {3} \ to \ mathbb {R}}$

For a homogeneous body, the mass density of which has the value everywhere in its interior , the total mass is the product of density and volume , i. i.e., it applies ${\ displaystyle \ rho _ {0}}$${\ displaystyle m}$${\ displaystyle V}$

${\ displaystyle m = \ rho _ {0} \, V}$.

In the case of inhomogeneous bodies , the total mass is more generally the volume integral

${\ displaystyle m = \ int _ {V} \ rho (\ mathbf {x}) \, \ mathrm {d} V}$

about the mass density.

The density results from the masses of the atoms that make up the material and from their spacing. In a homogeneous material, for example in a crystal, the density is the same everywhere. It usually changes with temperature and, in the case of compressible materials (such as gases), with pressure . Therefore, for example, the density of the atmosphere is location-dependent and decreases with altitude.

The reciprocal value of the density is called the specific volume and plays a role primarily in the thermodynamics of gases and vapors . The ratio of the density of a substance to the density under normal conditions is called the relative density .

In the first edition of DIN 1306 density and weight; Terms from August 1938, the density in today's sense was standardized as mean density and the location-dependent density at a point was simply defined as density : “The density (without the addition of 'mean') in a point of a body is the limit value to which the mean density tends towards a volume containing the point, if one thinks this reduced so much that it becomes small compared to the dimensions of the body, but still remains large compared to the structural units of its substance. ”In the August 1958 issue, the mean density was then expressed in density renamed with the explanation: "Mass, weight and volume are determined on a body whose dimensions are large compared to its structural components."

## Examples

The density of individual substances and materials can be found on the respective Wikipedia page. The density of elementary substances is also given in the list of chemical elements .

material Mass density
Interstellar matter 10 2 ... 10 9  atoms / m 3  ≈ 10 −13 ... 10 −6  g / km 3
Gases 0.09 kg / m 3  ( hydrogen ) ... 0.18 kg / m 3  ( helium ) ... 1.29 kg / m 3  ( air ) ... 1.78 kg / m 3  ( argon ) ... 12.4 kg / m 3  ( tungsten (VI) fluoride )
Wood 200 kg / m 3 … 1,200 kg / m 3
liquids 616 kg / m 3 ( isopentane ) ... 1000 kg / m 3 ( water ) ... 1 834 kg / m 3 ( sulfuric acid H 2 SO 4 ) ... 3 119 kg / m 3 ( bromine Br 2 ) ... (various heavy liquids ) ... 13 595 kg / m 3 ( mercury Hg)
Metals 534 kg / m 3  ( lithium )… 7874 kg / m 3 ( iron Fe)… 19 302 kg / m 3  ( gold )… 22 590 kg / m 3  ( osmium )
concrete 800 to 2 000 kg / m 3  ( lightweight concrete ) ... 2 400 kg / m 3  (normal concrete ) ... 2 600 to 4 500 kg / m 3  ( heavy concrete )
Stars 1 400 kg / m 3  (our sun ) ... 1.1 · 10 6  kg / m 3  (stars with helium burning ) ... 10 13  kg / m 3  ( nuclear fusion of heavy elements) ... 10 17 to 2.5 · 10 18  kg / m 3 ( neutron star )

## Sealing powders and porous materials

In the case of porous materials, a distinction has to be made between the skeletal or true density , where the mass is related to the volume without the pores, and the apparent density , which refers to the total volume including the pores. In the case of powders , bulk solids and heaps , the apparent density also depends on whether the material was heaped up loosely or tamped. Accordingly, a distinction is made between the bulk density and the vibration or tamped density . The relationship between bulk volume and tamped volume is also called the Hausner factor .

## Expansion coefficient

The change in environmental conditions leads to a change in density. The expansion coefficient is generally not constant, but depends on the ambient conditions, for example the temperature. For two temperatures and with , a mean statistical volume expansion coefficient can be calculated, from which the quotient of the two densities and can be calculated: ${\ displaystyle T_ {0}}$${\ displaystyle T_ {2}}$${\ displaystyle T_ {2}> T_ {0}}$${\ displaystyle \ gamma _ {\ text {avg.}}}$${\ displaystyle \ rho _ {0}}$${\ displaystyle \ rho _ {2}}$

${\ displaystyle {\ frac {\ rho _ {0}} {\ rho _ {2}}} = 1+ \ gamma _ {\ text {middle}} \ cdot (T_ {2} -T_ {0}) }$