Density: The Platinum Flute

Edgard Varese

Edgard Victor Achille Charles Varèse (December 22, 1883 – November 6, 1965) was an innovative French-born composer who spent the greater part of his career in the United States. Varèse's music features an emphasis on timbre and rhythm. He was the inventor of the term "organized sound", a phrase meaning that certain timbres and rhythms can be grouped together, sublimating into a whole new definition of sound. His use of new instruments and electronic resources led to his being known as the "Father of Electronic Music" while Henry Miller described him as "The stratospheric Colossus of Sound".

Edgar Varese composed a piece called "Density 21.5"^[1] for the inaugural performance of a platinum flute. Several performances are on Youtube Platinum is very "heavy", with a density of 21.5 g/cm³, so the flute must have been significantly heavier than one made of silver. Platinum is the heaviest element except for osmium (22.61 g/cm³.

The terms heavy and light are commonly used in two different ways. We refer to weight when we say that an adult is heavier than a child. On the other hand, something else is alluded to when we say that platinum is heavier than silver. A small platinum ring would obviously weigh less than a silver flute, but platinum is heavier in the sense that a piece of given size weighs more than the same-size piece of silver.

What we are actually comparing is the mass per unit volume, that is, the density. In order to determine these densities, we might weigh a cubic centimeter of each type of metal. If the platinum sample weighed 21.45 g and the silver 10.49 g, we could describe the density of platinum as 21.45 g cm^–3 and that of silver as 10.49 g cm^–3. (Note that the negative exponent in the units cubic centimeters indicates a reciprocal. Thus 1 cm^–3 = 1/cm³ and the units for our densities could be written as \(\dfrac{\text{g}}{\text{cm}^3}\), g/cm³, or g cm^–3. In each case the units are read as grams per cubic centimeter, the per indicating division.) We often abbreviate "cm³" as "cc", and 1 cm³ = 1 mL exactly by definition.

In general it is not necessary to weigh exactly 1 cm³ of a material in order to determine its density. We simply measure mass and volume and divide volume into mass:

\[\text{Density} = \dfrac{\text{mass}} {\text{volume}}\]

or

\[\rho = \dfrac{\text{m}} {\text{V}}\]

where

• ρ = density
• m = mass
• V = volume

Densities of Common Elements and Compounds

Substance	Density/g/cm^3 at 20^oC
Helium gas	0.00018
Air	0.00128
styrofoam	0.03 - 0.12
cork	0.22- 0.26
Argon	0.0018
gasoline	.66 - .69
grain alcohol	0.79
plastics	0.85 - 1.4
Water	1.00
Aluminum	2.7
iron	7.87
Gold	17.31
platinum	21.45
silver	10.49
osmium	22.61
Densities of many more materials are easily found.

Example \(\PageIndex{1}\): Density Calculation

Calculate the density of (a) a piece of platinum whose mass is 298.155 g and which, when submerged, increases the water level in a graduated cylinder by 13.90 ml; (b) a platinum cylinder of mass 199.029 g, radius 0.750 cm, and height 5.25 cm.

Solution

a) Since the submerged metal displaces its own volume,

\(\text{Density} =\rho = \dfrac{\text{m}} {\text{V}} = \dfrac{298.155 g} {13.9 mL} = \text {21.45 g/mL or 21.45 g mL}^{-1}\)

b) The volume of the cylinder must be calculated first, using the formula

\(\text{V} = {\pi} r^{2} h = 3.142 × \text{(0.750 cm)}^{2} * 5.25 \text{cm} = \text{9.278 718 8 cm}^{3}\)

Then \(\rho = \dfrac{\text{m}} {\text{V}} = \dfrac{199.029g} {9.278 718 8cm^{3}}\)

=\(\text{21.5} \dfrac{\text{g}}{\text{cm}^3}\)

Note

Note that unlike mass or volume, the density of a substance is independent of the size of the sample. Thus density is a property by which one substance can be distinguished from another. A sample of pure platinum can be trimmed to any desired volume or adjusted to have any mass we choose, but its density will always be 22.50 g/cm³ at 20°C. The densities of some common pure substances are listed in the Table.

Tables and graphs are designed to provide a maximum of information in a minimum of space. When a physical quantity (number × units) is involved, it is wasteful to keep repeating the same units. Therefore it is conventional to use pure numbers in a table or along the axes of a graph. A pure number can be obtained from a quantity if we divide by appropriate units. For example, when divided by the units gram per cubic centimeter, the density of aluminum becomes a pure number 2.70:

\[\dfrac{\text{Density of aluminum}} {\text{1 g cm}^{-3}} = \dfrac{\text{2.70 g cm}^{-3}} {\text{1 g cm}^{-3}} = 2.70\]

Therefore, a column in a table or the axis of a graph is conveniently labeled in the following form:

\[\dfrac{\text{Quantity}}{\text{units}}\]

This indicates the units that must be divided into the quantity to yield the pure number in the table or on the axis. This has been done in the second column of the Table.

Converting Densities

In our exploration of Density, notice that chemists may express densities differently depending on the subject. The density of pure substances may be expressed in kg/m³ in some journals which insist on strict compliance with SI units; densities of soils may be expressed in lb/ft³ in some agricultural or geological tables; the density of a cell may be expressed in mg/µL; and other units are in common use. It is easy to transform densities from one set of units to another, by multiplying the original quantity by one or more unity factors:

Example \(\PageIndex{2}\): Density Conversion

Convert the density of water, 1 g/cm³ to (a) lb/cm³ and (b) lb/ft³

a. The equality 454 g = 1 lb can be used to write two unity factors,

\(\dfrac{\text{454 g}} {\text{1 lb}}\) or \(\dfrac{\text{1 lb}} {\text{454}}\)

The given density can be multiplied by one of the unity factors to get the desired result. The correct conversion factor is chosen so that the units cancel:

\(\dfrac{\text{1 g}} {\text{cm}^{3}}* \dfrac{\text{1 lb}} {\text{454 g}} = 0.002203 \dfrac{\text{lb}} {\text{cm}^{3}}\)

b. Similarly, the equalities 2.54 cm = 1 inch, and 12 inches = 1 ft can be use to write the unity factors:

\(\dfrac{\text{2.54 cm}} {\text{1 in}}\), \(\dfrac{\text{1 in}} {\text{2.54 cm}}\), \(\dfrac{\text{12 in}} {\text{1 ft}}\) and \(\dfrac{\text{1 ft}} {\text{12 in}}\)

In order to convert the cm³ in the denominator of 0.002203 \(\dfrac{lb} {cm^{3}}\) to in³, we need to multiply by the appropriate unity factor three times, or by the cube of the unity factor:

\(\text{0.002203} \dfrac{\text{g}} {\text{cm}^{3}}\) x \(\dfrac{\text{2.54 cm}} {\text{1 in}}\) x \(\dfrac{\text{2.54 cm}} {\text{1 in}}\) x \(\dfrac{\text{2.54 cm}} {\text{1 in}}\)

or

\(\text{0.002203} \dfrac{\text{g}} {\text{cm}^{3}}\) x \((\dfrac{\text{2.54 cm}} {\text{1 in}})^{3} = \text{0.0361} \dfrac{\text{lb}}{\text{in}^3}\)

This can then be converted to lb/ft³:

\(\text{0.0361} \dfrac{\text{lb}} {\text{in}^{3}}\) x \((\dfrac{\text{12 in}} {\text{1 ft}})^{3} = \text{62.4} \dfrac{\text{lb}}{\text{ft}^3}\)

Note

It is important to notice that we have used conversion factors to convert from one unit to another unit of the same parameter

From ChemPRIME: 1.8: Density

Contributors and Attributions

• Ed Vitz (Kutztown University), John W. Moore (UW-Madison), Justin Shorb (Hope College), Xavier Prat-Resina (University of Minnesota Rochester), Tim Wendorff, and Adam Hahn.