Conversion Factors and Fitness Testing

It's probably no surprise that if your weight is more than optimal for your height, your belt may feel a little tight. But we can calculate the increase in size (volume) of body tissue when Fat Free body Mass (FFM) is converted to the same mass of fat.

Earlier we showed how unity factors can be used to express quantities in different units of the same parameter. For example, a density can be expressed in g/cm³ or lb/ft³. Now we will also how conversion factors representing mathematical functions, like D = m/v, can be used to transform quantities into different parameters. Unity factors and conversion factors are conceptually different, and we'll see that the "dimensional analysis" we develop for unit conversion problems must be used with care in the case of functions.

Body Composition

Body composition measures are important to optimize competitive performance and assess health.

Body composition can be determined by underwater weighing (densitometry), where the subject is weighed in a chair that is lowered into water. The subject exhales completely, and the weight is corrected for residual lung volume. The density is the ratio

\[\text{D} = \dfrac{\text{mass in air}}{\text{mass* in air - mass* in water}}\]

*The mass of the sample does not change, so it would be more appropriate to say weight. Here "mass" should be understood as "mass equivalent" of the weight, since balances are calibrated in grams, a unit of mass. Weights are technically measured in Newtons. Also, this equation is dimensionless (the unit g in the numerator and denominator cancel), so it is technically not a density, but a "specific gravity". Physiologists use it under conditions where it gives density values, so we've symbolized it D, not ρ

The denominator represents the volume, because it is the bouyancy, or the mass of water displaced. Near room temperature where the density of water is ~1, the mass of water displaced is numerically equal to the volume of water (V = m/D = m/1), which is equal to the volume of the body.

Because whole body densitometry is so difficult, a variety of shortcuts for fitness testing have been devised. For example, body fat can be estimated by skinfold thickness measurement, The skin is pinched at the triceps to raise double layer of skin and the underlying adipose tissue, but not the muscle. Calipers are then applied 1 cm below and at right angles to the pinch, and a reading in millimeters (mm) taken two seconds later. The mean of two measurements should be taken. If the two measurements differ greatly, a third should then be done, then the median value taken. This measurement is used in combination with measurements of the waist, and age in the equations below ^[1]

Percent BF for men = 0.353 waist (cm) + 0.756 triceps (mm) + 0.235 age (y) - 26.4; for women = 0.232 waist (cm) + 0.657 triceps (mm) + 0.215 age (y) - 5.5. [2]. All formulas for BF may be inaccurate because they fail to account for "frame size", or skeletal density, which is 3.317 g cm^-3.

Effect of BF on Body Size

To get a "feel" for the difference between muscle and fat tissue, suppose we think of a cylinder with a circumference of 27 inches (like some people's waistlines) and height of 2 feet. This might be a simple model of the "trunk" or torso of our body. The density of fat (adipose tissue) is generally about 0.901 ^[2] while the density of FFM (fat free mass) is in the range 1.075 to 1.127 g/cm³.

It's interesting to do some calculations on our model torso. The volume of our model torso in cm³ is calculated as follows:

\[\text{27 in} \times \dfrac{\text{2.54 cm}}{\text{in}} = \text{68.6 cm}\]

(similarly for h = 2 feet)

\[\text{C} = \text{2} \pi \text{r}\]

\[\text{V} = \pi \text{r}^2 \text{h}\]

\[= \pi (\dfrac{\text{C}}{\text{2}\pi})^2 \text{h} = \text{22815 cm}^3\]

Mass from Volume

Now, we can calculate the mass of the model torso made out of FFM whose density is 1.12 g/cm³ from the mathematical function which defines density:

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

or

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

If we multiply both sides by V, we obtain

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

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

So for a volume of 22 815 cm³ and ρ = 1.12 g/cm³, we calculate a mass of

\[\text{m} = \text{22 815 cm}^3 \times \text{1.12} \dfrac{\text{g}}{\text{cm}^3} = \text{25 554 g}\]

or in pounds

\[\text{25 554 g} \times \dfrac{\text{1 lb}}{\text{453.59 g}} = \text{56.3 lb}\]

Volume from Mass

The formula which defines density can also be used to convert the mass of a sample to the corresponding volume. If both sides of Equation (1.2) are multiplied by 1/ρ, we have

\[\dfrac{1}{\rho} \times \text{m} = \text{V} \rho \times \dfrac{1}{\rho} = \text{V}\]

\[\text{V} = \text{m} \times \dfrac{1}{\rho}\]

Suppose our model torso is now converted to fat, with a density of 0.901 g/cm³. What is the new circumference of the "torso"?

\[\text{V} = \dfrac{\text{m}}{\text{D}} = \dfrac{\text{25 554 g}}{\text{0.901}} \dfrac{g}{cm^3} = \text{28 362 cm}^3\]

and, keeping the height = 24 in, the circumference is calculated from the formula above:

\[\text{V} = \text{28 362 cm}^3 = \pi (\dfrac{\text{C}}{\text{2}\pi})^2 \text{h}\]

C = 76.5 cm or 30.0 in. The belt gained about 3", for the same mass converted from FFM to adipose.

Conversion Factors vs. Unity Factors

Notice that we used the mathematical function D = m/V to convert parameters from mass to volume or vice versa in these examples. How does this differ from the use of unity factors to change units of one parameter?

An Important Caveat

A mistake sometimes made by beginning students is to confuse density with concentration, which also may have units of g/cm³. By dimensional analysis, this looks perfectly fine. To see the error, we must understand the meaning of the function

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

In this case, V refers to the volume of a solution, which contains both a solute and solvent.

Given a concentration of an alloy is 10 g gold in 100 cm³ of alloy, we see that it is wrong (although dimensionally correct as far as conversion factors go) to incorrectly calculate the volume of gold in 20 g of the alloy as follows:

\[\text{20 g} \times \dfrac{\text{100 cm}^3}{\text{10 g}} = \text{200 cm}^3\]

It is only possible to calculate the volume of gold if the density of the alloy is known, so that the volume of alloy represented by the 20 g could be calculated. This volume multiplied by the concentration gives the mass of gold, which then can be converted to a volume with the density function.

The bottom line is that using a simple unit cancellation method does not always lead to the expected results, unless the mathematical function on which the conversion factor is based is fully understood.

Example \(\PageIndex{1}\): Volume of Ethanol

A solution of ethanol with a concentration of 0.1754 g / cm³ has a density of 0.96923 g / cm³ and a freezing point of -9 ° F ^[3]. What is the volume of ethanol (D = 0.78522 g / cm³ at 25 °C) in 100 g of the solution?

Solution

The volume of 100 g of solution is

\[\text{V} = \dfrac{m}{D} = \dfrac{\text{100 g}}{\text{0.96923} \dfrac{g}{\text{cm}^3}} = \text{103.17 cm}^3\]

The mass of ethanol in this volume is

\[\text{m} = \text{V} \times \text{C} = \text{103.17 cm}^3 \times \text{0.1754} \dfrac{\text{g}}{\text{cm}^3} = \text{18.097 g}\]

The volume of ethanol =

\[\dfrac{m}{} = \dfrac{\text{18.097 g}}{\text{0.78522} \dfrac{g}{cm}} = \text{23.05 cm}\]

Note that we cannot calculate the volume of ethanol by

\[\dfrac{\dfrac{0.96923 g}{\text{cm}^3} \times \text{100cm}^3}{\dfrac{\text{0.78522 g}}{\text{cm}^3}} = \text{123.4 cm}^3\]

even though this equation is dimensionally correct.

Note

Note that this result required when to use the function C = m/V, and when to use the function D=m/V as conversion factors. Pure dimensional analysis could not reliably give the answer, since both functions have the same dimensions.

The two calculations just done show that density is a conversion factor which changes volume to mass, and the reciprocal of density is a conversion factor changing mass into volume. This can be done because of the mathematical formula, D = m/v, which relates density, mass, and volume. Algebraic manipulation of this formula gave us expressions for mass and for volume, and we used them to solve our problems. If we understand the function D = m/V and heed the caveat above, we can devise appropriate conversion factors by unit cancellation, as the following example shows:

Example \[\PageIndex{2}\]: Volume of Mercury

A student weighs 98.0 g of mercury. If the density of mercury is 13.6 g/cm³, what volume does the sample occupy?

Solution:

We know that volume is related to mass through density.

Therefore

\[\text{V} = \text{m} \times \text{conversion factor}\]

Since the mass is in grams, we need to get rid of these units and replace them with volume units. This can be done if the reciprocal of the density is used as a conversion factor. This puts grams in the denominator so that these units cancel:

\[\text{V} = \text{m} \times \dfrac{1}{\rho} = \text{98.0 g} \times \dfrac{\text{1 cm}^3}{\text{13.6 g}} = \text{7.21 cm}^3\]

If we had multiplied by the density instead of its reciprocal, the units of the result would immediately show our error:

\[\text{V} = \text{98.0 g} \times \dfrac{\text{13.6 g}}{\text{1 cm}^3} = \text{1.333} \dfrac{\text{g}^2}{\text{cm}^3}\]
(no cancellation!)

It is clear that square grams per cubic centimeter are not the units we want.

Using a conversion factor is very similar to using a unity factor—we know the factor is correct when units cancel appropriately. A conversion factor is not unity, however. Rather it is a physical quantity (or the reciprocal of a physical quantity) which is related to the two other quantities we are interconverting. The conversion factor works because of that relationship [Eqs. (1.1), (1.2), and (1.3) in the case of density, mass, and volume], not because it is equal to one. Once we have established that a relationship exists, it is no longer necessary to memorize a mathematical formula. The units tell us whether to use the conversion factor or its reciprocal. Without such a relationship, however, mere cancellation of units does not guarantee that we are doing the right thing.

A simple way to remember relationships among quantities and conversion factors is a “road map“of the type shown below:

\[\text{Mass }\overset{density}{\longleftrightarrow}\text{ volume or }m\overset{\rho }{\longleftrightarrow}V\text{ }\]

This indicates that the mass of a particular sample of matter is related to its volume (and the volume to its mass) through the conversion factor, density. The double arrow indicates that a conversion may be made in either direction, provided the units of the conversion factor cancel those of the quantity which was known initially. In general the road map can be written

\[\text{First quantity }\overset{\text{conversion factor}}{\longleftrightarrow}\text{ second quantity}\]

As we come to more complicated problems, where several steps are required to obtain a final result, such road maps will become more useful in charting a path to the solution.

Example \[\PageIndex{3}\]: Weight Calculation

Black ironwood has a density of 67.24 lb/ft³. If you had a sample whose volume was 47.3 ml, how many grams would it weigh? (1 lb = 454 g; 1 ft = 30.5 cm).

Solution

The road map

\[\text{m}\overset{\rho }{\longleftrightarrow}V\text{ }\nonumber\]

tells us that the mass of the sample may be obtained from its volume using the conversion factor, density. Since milliliters and cubic centimeters are the same, we use the SI units for our calculation:

\[\text{Mass} = \text{m} =\text{47.3 cm}^3 \times \dfrac{\text{67.24 lb}}{\text{1 ft}^3}\]

Since the volume units are different, we need a unity factor to get them to cancel:

\[\text{m} = \text{47.3 cm}^3 \times (\dfrac{\text{1 ft}}{\text{30.5 cm}})^3 \times \dfrac{\text{67.24 lb}}{\text{1 ft}^3} = \text{47.3 cm}^3 \times \dfrac{\text{1 ft}^3}{\text{30.5}^3 \text{cm}^3} \times \dfrac{\text{67.24 lb}}{\text{1 ft}^3}\]

We now have the mass in pounds, but we want it in grams, so another unity factor is needed:

\[\text{m} = \text{47.3 cm}^3 \times \dfrac{\text{1 ft}^3}{\text{30.5}^3 \text{cm}^3} \times \dfrac{\text{67.24 lb}}{\text{1 ft}^3} \times \dfrac{\text{454 g}}{\text{1 lb}} = \text{50.9 g}\]

In subsequent chapters we will establish a number of relationships among physical quantities. Formulas will be given which define these relationships, but we do not advocate slavish memorization and manipulation of those formulas. Instead we recommend that you remember that a relationship exists, perhaps in terms of a road map, and then adjust the quantities involved so that the units cancel appropriately. Such an approach has the advantage that you can solve a wide variety of problems by using the same technique.

From ChemPRIME: 1.9:Conversion Factors and Functions

References

1. ↑ Michael EJ Lean, Thang S Han, and Paul Deurenberg A, J Clin Nutr l996;63:4-14 [1].
2. ↑ http://jap.physiology.org/cgi/reprint/90/4/1523
3. ↑ en.Wikipedia.org/wiki/Ethanol_(data_page)#Properties_of_aqueous_ethanol_solutions

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.