Density

This topic covers Density -- Claimed by Ritvik Khanna, M08 Dr. Greco Spring 2016

The Main Idea

Density can be simply defined as the relative compactness of a substance. The consistency of the matter at hand can be quantified by mass per unit volume. The density, also called the volumetric mass density, of a substance is denoted by the greek symbol ρ (the lowercase Greek letter rho) but sometimes it can also be represented by the Latin letter D. The density of materials varies, different materials will have different densities. For instance, osmium and iridium are different elements and therefore have different densities. Osmium has a density of 22.59 g/cm^3 while iridium has a density of 22.42 g/cm^3. Also the density of a material can be related to the materials buoyancy, packaging, and purity.

A Mathematical Model

Mathematically density is defined as mass divided by volume.

[math]\displaystyle{ \rho = \frac{m}{V}, }[/math]

Density = Mass / Volume

In this equation ρ is the density, m is the mass, and V is the volume.The mass can be expressed in grams [g] or kilograms [kg], and the volume is measured in liters [L], cubic centimeters [cm^3], or milliliters [mL]. In this equation mass is divided by volume and in the US oil and gas industry sometimes density is referred to as weight per unit volume. This is technically incorrect as this quantity is actually called specific weight.

A Computational Model

The density of a solution is the sum of the mass of the components of that solution. Mass concentrations of each component ρi in a solution sums to density of the solution. Therefore this can be modeled by the following equation.

\rho = \sum_i \varrho_i \,

This expressed as a function of the densities of pure components of the mixture allows for the computation of the excess molar volumes through the following summation.

\rho = \sum_i \rho_i \frac{V_i}{V}\,= \sum_i \rho_i \varphi_i =\sum_i \rho_i \frac{V_i}{\sum_i V_i + \sum_i V_i^{E}}

From this point by having the excess volumes and activity coefficients it is possible to determine the activity coefficients by using the equation below.

\bar{V^E}_i= RT \frac{\partial (ln(\gamma_i))}{\partial P}

Examples

Simple

What is the weight of the isopropyl alcohol that exactly fills a 200.0 mL container? The density of isopropyl alcohol is 0.786 g/mL.

d=g/mL

g=(d)(mL)

g=(0.789 g/mL)(200.0 mL) = 157.2 g

...therefore the isopropyl alcohol weights [math]\displaystyle{ 157.2 }[/math] grams

Middling

A copper ingot has a mass of 2.15 kg. If the copper is drawn into wire whose diameter is 2.27 mm, how many inches of copper wire can be obtained from the ingot?

a) Determine volume of copper:

[math]\displaystyle{ 8.94 }[/math] g/cm^3 = [math]\displaystyle{ 2150 }[/math] g/x x = [math]\displaystyle{ 240.49217 }[/math] cm^3

Note: this is the volume of the wire.

b) Determine h in the volume of a cylinder (the wire)

[math]\displaystyle{ }[/math]

[math]\displaystyle{ 240.49217 }[/math] cm^3 = [math]\displaystyle{ 3.14159 }[/math] * [math]\displaystyle{ 0.1135 }[/math]*2 *{h}

{h} = [math]\displaystyle{ 5942.37 }[/math] cm

Note: 0.1135 cm is the radius

c) Convert cm to inch:

5942.37 cm divided by 2.54 cm/in = 2340 inches

Difficult

Copper can be drawn into thin wires. How many meters of 34-gauge wire (diameter =[math]\displaystyle{ 6.304 x 10^ {-3} }[/math] inches) can be produced from the copper that is in[math]\displaystyle{ 5.88 }[/math] pounds of covellite, an ore of copper that is [math]\displaystyle{ 66% }[/math] copper by mass? (Hint: treat the wire as a cylinder. The density of copper is [math]\displaystyle{ 8.94 }[/math]g/ cm^3; one kg weighs [math]\displaystyle{ 2.2046 }[/math] lb. The volume of a cylinder is πr2h.

a) Determine pounds of pure copper in 5.88 lbs of covellite:

5.88 lbs x 0.66 = 3.8808 lbs

b) Convert pounds to kilograms:

3.8808 lbs ÷ 2.6046 lbs kg¯1 = 1.489979 kg (I'm keeping a few guard digits) 1.489979 kg = 1489.979 g

c) Determine volume this amount of copper occupies:

8.94 g cm¯3 = 1489.979 g / x x = 166.664 cm3

Note: this is the volume of the cylinder.

d) Convert the diameter in inches to a radius in centimeters:

dia = 6.304 x 10¯3 in; radius = 3.152 x 10¯3 in (3.152 x 10¯3 inch) (2.54 cm / inch) = 8.00 x 10¯3 cm

e) Determine h in the volume of a cylinder:

166.664 cm3 = (3.14159) (8.00 x 10¯3 cm)2 h h = 8.29 x 105 cm = 5.63 x 103 m

Connectedness

Density is important in construction of biomaterials that could potentially have clinical applications.

History

Archimedes in 250 B.C. "discovered" density. He suspected that he was being cheated on by the metal craftsmen who had constructed his golden crown. Archimedes suspected that the craftsman was using a sizeable amount of silver with his gold crown, which is cheaper, and ultimately an insult to the king. Archimedes put the crown that the craftsmen had created as well as a crown of pure gold both into a bathtub full of water. He measured the amount of water that was in excess and confirmed his suspicions. He then proceeds to chant "Eureka! Eureka!" in the streets.

Chabay & Sherwood: Matters and Interactions -- Modern Mechanics Volume 1, 4th Edition

Density

Contents

The Main Idea

A Mathematical Model

A Computational Model

Examples

Simple

Middling

Difficult

Connectedness

History

See also

Further reading

External links

References

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