Experiment #1

Fluid Properties: Density & Surface Tension

Stephen Mirdo

Performed on September 23, 2010

Report due September 30, 2010

Table of Contents Object ………………………………………..………………………….………….…. p. 1 Theory …………………………………………………………………………..…pp. 1 - 3 Procedure ……………………………………………………………………..…...pp. 4 - 6 Results ………………………………………………………………...……..……pp. 7 - 8 Discussion and Conclusion …………………………………………………..…...pp.9 - 10 Appendix ……………………………………………………..…….……..……pp. 11 - 13

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Object

The object of this fluid property experiment was to determine the density of a fluid and use this determined density to calculate the fluid’s surface tension.

Theory

The density of a fluid is defined as its mass per unit volume and is a function of

pressure and temperature. If a fluid undergoes thermal expansion, its molecules will be distributed more widely and will therefore be less dense. There will be fewer molecules, thus less mass, per a given unit of volume. Likewise, if a fluid experiences thermal contraction due to cooling, the kinetic energy of the molecules in the fluid will be a lesser value and they will be packed closer together. In this case, the fluid will have a higher mass per unit volume, therefore a higher density than the former case.

ρ = m/V (Equation 1)

Pressure likewise affects the density of a fluid. Consider a fluid with a free surface exposed to the atmosphere. If the pressure on the surface of the fluid were to increase, the density of the fluid would increase due to being compressed. The converse of this scenario is also true: a decrease in the pressure on the free surface of the fluid would decrease its density.

The density of a fluid can be obtained through various means. One method to

determine a fluid’s density is to weigh a known volume and divide its mass by the volume. To perform this type of analysis, weigh a fluid vessel such as graduated beaker with and without the known volume of fluid as seen in Equation 2 below. Simply divide the determined mass by the volume as in Equation 1 to obtain the fluid’s density.

m laden beaker – m unladed beaker = m fluid (Equation 2)

Another method to determine a fluid’s density is to calculate the buoyant force it exerts on a suspended object and divide this force by the known volume of the fluid. Use a hanging scale to suspend an object and record its apparent weight. Then, weigh the mass again, this time submerged in a known volume of fluid. Calculate the buoyant force by calculating the difference between these two values as seen in Equation 3. Divide the buoyant force by the volume the object displaced to determine the density (Equation 4).

Buoyant Force = Weight dry – Weight submerged (Equation 3)

ρ = B / V (Equation 4)

The last method for measuring density to be considered in this report involves the use of a hydrometer cylinder. A hydrometer is an instrument used to measure the specific gravity of a fluid, usually with a reference to pure water at room temperature. This

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means that the specific gravity of a fluid is the ratio of the mass of a liquid to the mass of an equal volume of pure water. To calculate the density of a fluid with this instrument, suspend the hydrometer bulb end down in a cylinder filled with fluid and wait for it to come to a rest. It is important that the hydrometer does not touch the sides of the cylinder so that the fluids other properties do not interfere with the reading. On the length of the hydrometer there are printed calibration marks. The value that is at the meniscus of the fluid should be recorded and is denoted as s. To calculate the density from this unitless value, multiply the hydrometer reading by the known density of water in the unit system of choice (1000 kg/m3 for SI and 62.4 lbm/ft3 for BG), as seen in Equation 5.

ρ = s * Volume of Water (Equation 5)

Another important fluid property to be considered in this experiment is surface

tension. The surface tension is defined as the measure of cohesive energy present at the interface of the fluid, or the amount of work required to extend the surface of a liquid. The units for surface tension are a force per unit length.

σ = F/L (Equation 6)

The fluid molecules beneath an interface experience a larger net intermolecular

attractive force. A molecule beneath the surface experiences attractive forces on all sides, thus experiencing a net force of zero and residing in a lower energy state. The fluid molecules at the surface, having no neighbors above them, are subjected to the attractive force of the molecules beneath them and are pulled inward towards the bulk of the fluid. This inward force causes the fluid to contract to a minimal area. This property explains why free falling fluids tend to take a spherical shape: a sphere has the lowest ratio of surface area to volume than any other three dimensional shape.

A method of obtaining the surface tension of a fluid is to employ a surface tension

meter. This device will measure the amount of energy per unit area requires to increase the surface area of a fluid. It does this by putting a wire inside the device under torsion, which in turn will raise a ring suspended in the fluid to and through the surface. Once it has broken free, the machine will stop recording and the displayed value is the apparent surface tension. The diagram below shows the general setup for a surface tension meter.

Figure 1: A schematic of a surface tension meter (Source: A Manual for the Fluid Mechanics Laboratory, William. S. Janna)

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The apparent surface tension recorded by the machine, measured in dynes per centimeter, must undergo correction to obtain the actual value for the surface tension of the fluid. To obtain the correction factor, evaluate Equation 7 below where σa is the apparent surface tension, ρ is the density of the fluid and r/R is a given property of the ring that breaks the fluid’s surface. The actual surface tension of the fluid is calculated by multiplying the apparent surface tension by this correction factor (Equation 8). _________________________________

F = 0.725 + √0.0004033(σa/ρ) + 0.04534 – 1.679(r/R) (Equation 7)

σ = F*σa (Equation 8)

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Procedure Part I: Determining the Density of a Fluid Equipment:

Graduated Cylinder 250 mL Graduated Beaker Suave Ocean Breeze Shampoo Fisher Hydrometer Cylinder Model 11-583-D Mettler Toledo PB153 S/Fact Scale (SN: CV3279) Crystal Hanging Mass

Method One:

1) Weigh the empty graduated beaker on a scale and record the mass. 2) Remove the beaker from the scale and measure an exact volume of 5 mL of Suave

Ocean Breeze shampoo into the beaker. 3) Weigh the laden beaker on the scale again and record the mass. 4) Determine the difference in the masses as per Equation 2 and record this mass.

This value is the mass of the fluid. 5) Divide the mass of the fluid by the volume measured into the beaker as per

Equation 3 and record this value. This will yield the density of the fluid. Method Two:

1) Suspend the crystal hanging mass from a harness attached directly to the plate of the Mettler Toledo PB 153 (SN: CV3279) and record the crystal’s apparent weight.

2) Fill a graduated beaker with an exact amount of 150 mL Suave Ocean Breeze shampoo and record this volume on a data sheet.

3) Remove the crystal hanging mass from the harness on the plate of the scale. Place the fluid filled beaker onto a riser over the plate so the weight of the fluid and beaker are neglected. Hang the crystal mass from the harness again, this time submerging the crystal completely in the fluid as seen in Figure 2 below. Record the weight of the mass as is it submerged in the fluid.

Figure 2: Schematic of hanging mass submerged in the known volume of fluid.

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4) Before removing the hanging mass from the fluid, record the volume of fluid the hanging mass displaced inside the beaker.

5) Determine the buoyant force exerted by the fluid on the hanging mass by subtracting the submerged weight from the non-submerged weight as per Equation 3 and record this value.

6) Divide the buoyant force by buoyant force calculated in Step 5 by the volume determined in Step 4 to determine the fluid’s density, as per Equation 4.

Method Three:

1) Select a hydrometer cylinder via an educated guess of the supposed density of the fluid.

2) Fill a graduated cylinder with an arbitrary amount of Suave Ocean Breeze shampoo. The particular volume for this part of the experiment is inconsequential as long as there is enough to submerge the hydrometer to obtain an accurate reading.

3) Insert the hydrometer into the shampoo filled cylinder as close to the center as possible. If the hydrometer sticks to the walls of the cylinder, it will yield an inaccurate specific gravity for the fluid.

4) Once the hydrometer has stabilized and is no longer bobbing in the fluid, take a reading of the specific gravity on the neck of the hydrometer at the meniscus of the fluid as seen in Figure 3 and record this value.

Figure 3: Demonstration of proper hydrometer reading. (Source: The Louisiana Universities Marine Consortium)

5) Multiply the specific gravity of the fluid obtained in Step 4 by the density of

water (1000 kg/m3 for SI and 62.4 lbm/ft3 for BG) as per Equation 5. This calculation yields the density of the fluid.

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Part II: Determining the Surface Tension of Fluid Equipment:

Fisher Scientific Tensiomat Model 21 (SN: 84858) Platinum-Iridium Ring (SN: 576830-1) Small Beaker Suave Ocean Breeze Shampoo

1) Fill a small beaker with an arbitrary amount of Suave Ocean Breeze shampoo. The amount is inconsequential as long as there is enough fluid for the platinum-iridium ring to be inserted into the beaker beneath the free surface and not touch the bottom.

2) Insert the platinum-iridium ring beneath the surface of the fluid at an angle so as to not disturb the surface too much. Any breakage of the surface will skew the results of the surface tension meter.

3) Place the beaker with ring onto the movable table of the Tensiomat and affix the ring to the balance rod of the device.

4) Level the balance rod of the Tensiomat through careful movements of the tension knob on the right side of the device. Zero out the reading dial on the front of the machine.

5) Flip the up/down switch on the front of the Tensiomat to the up position. The device will put the internal wire of the machine through torsion and lift the ring to and through the surface of the fluid.

6) Once the ring has broken the surface of the fluid, the machine will disengage. The dial on the front of the Tensiomat will display the amount of force per unit length required to break the surface of the fluid in dynes per centimeter. This value is the apparent surface tension of the fluid.

7) Remove the ring from the beaker and administer a small amount of fluid to restore the surface. Repeat Steps 2 through 6 three more times and record the results.

8) To obtain the actual surface tension of the fluid, assess the correction factor by evaluating Equation 7 using the density acquired in Part 1 of this experiment and the r/R value for the platinum-iridium ring. Multiply this value by the apparent surface tension as per Equation 8 to obtain the actual surface tension of the fluid.

Figure 4: Tensiomat Model 21 Diagram

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Results Part I: Density Results Method One

Table 1: Measurements for Method One for Density Mass of Empty Beaker 26.71 g

Mass of Laden Beaker 31.87 g Volume in Beaker 5.0 mL Mass of Fluid in Beaker 5.16 g

Calculations for Method One

Mass of Fluid (m): 31.87 g – 26.71 g = 5.16 g Equation 2

Density of Fluid (ρ): 5.16 g / 5.0 cm3 = 1.032 g/cm3 = 1032 kg/m3 Equation 1

Method Two

Table 2: Measurements for Method Two for Density Dry Weight of Crystal 30.492 g Submerged Weight of Crystal 20.390 g

Volume in Beaker Before Submerging Crystal 150 mL

Volume in Beaker After Submerging Crystal 160 mL

Volume of Crystal 10 mL

Calculations for Method Two for Density

Buoyant Force (B): 30.492 g – 20.390 g = 10.102 g Equation 3

Density of Fluid (ρ): 10.102 g / 10 cm3 = 1.0102 g/cm3 = 1010 kg/m3 Equation 4

Method Three

Table 3: Measurements for Method Three for Density Hydrometer Cylinder Reading 1.014

Calculations for Method Three for Density

Density of Fluid (ρ): 1.014 * 1 g/cm3 = 1.014 g/cm3 = 1014 kg/m3 Equation 5

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Part II: Surface Tension Results

Table 4: Apparent Surface Tension and Calculation for Actual Surface Tension

r/R for Platinum-Iridium Ring: 1/53.5932016

Trial No. σa F σ = F* σa 1 32.5 dyne/cm 0.8891 28.9 dyne/cm 2 36.0 dyne/cm 0.8933 32.2 dyne/cm 3 28.0 dyne/cm 0.8836 24.7 dyne/cm 4 34.5 dyne/cm 0.8915 30.8 dyne/cm

Surface Tension = σ (avg) = 29.1 dyne/cm

Calculations for Surface Tension

Correction Factor (F) (Equation 7):

- Note: The hydrometer cylinder’s measurement will be used for the density for this equation because it is more accurate.

______________________________________________________________ 0.725 + √0.0004033[(32.5 dyne/cm)/(1.014 g/cm3)] + 0.04534 – 1.679(1/53.5932016) =

0.8891 ______________________________________________________________

0.725 + √0.0004033[(36.0 dyne/cm)/(1.014 g/cm3)] + 0.04534 – 1.679(1/53.5932016) = 0.8933

______________________________________________________________ 0.725 + √0.0004033[(28.0 dyne/cm)/(1.014 g/cm3)] + 0.04534 – 1.679(1/53.5932016) =

0.8836 ______________________________________________________________

0.725 + √0.0004033[(34.5 dyne/cm)/(1.014 g/cm3)] + 0.04534 – 1.679(1/53.5932016) = 0.8915

Actual Surface Tension (σ) (Equation 8):

0.8891 * 32.5 dyne/cm = 28.9 dyne/cm

0.8933 * 36.0 dyne/cm = 32.2 dyne/cm

0.8836 * 28.0 dyne/cm = 24.7 dyne/cm

0.8915 * 34.5 dyne/cm = 30.8 dyne/cm

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Discussion & Conclusion

The results for the density measurement portion of the experiment are not all the same value. However, they are in agreement with one another because they are not spread too far apart. If, for instance, one method for calculating the density of the shampoo yielded a value that was different by a factor of two, there would be cause for concern regarding the accuracy of the method. Because the calculated values for all three methods fall within a small range, it can be concluded that these values are in agreement.

The apparent buoyant force in method two for density measurement would vary with respect to depth. Assuming the fluid is incompressible and the object submerged in the fluid is also incompressible, the actual buoyant force would remain constant no matter to what depth the object is submerged. Because of the method employed in this experiment, where the buoyant force is determined by the difference in apparent weight dry and submerged, the buoyant force acting on the crystal would vary. This variance is the result of the pressure the column of fluid exerts on the submerged object. At an increasing depth of the object, there will be more fluid in the column above and therefore an increase in the object’s apparent weight. The result of this discrepancy is a decrease in the buoyant force with an increase in depth.

The most accurate result for the density of the Suave Ocean Breeze shampoo was determined by the hydrometer cylinder method. The hydrometer is a specially calibrated instrument with the purpose of measuring the specific gravity of a fluid. Because only one measurement from the device is required to determine the specific gravity of a fluid, it tends to produce a more accurate result for the density.

The values of 1032 kg/m3, 1010 kg/m3 and 1014 kg/m3 are very near one other. Each method employed to obtain the density of the Suave Ocean Breeze shampoo yielded results that were consistent with one another. Therefore, it can be concluded that these results are precise. Although they are consistent, these results cannot be considered to be accurate. The true value for the density of the fluid may be more or less than the results this experiment calculated.

The mean, or average value, of the density results is 1019 kg/m3. This value is obtained by dividing the sum of all the calculated densities by the number of methods employed. The mean value of the densities is most likely closer to the true value of the density of Suave Ocean Breeze Shampoo.

Mean = (1032 kg/m3 + 1010 kg/m3 + 1014 kg/m3) / 3 = 1019 kg/m3

The standard deviation of the density results is 11.65 kg/m3. This value is obtained by taking the square root of the quotient of the sum of the densities, less the mean, squared and divided by the number of methods minus one. ______________

σ = √ Σ (x-xavg)2 / (n-1)

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The standard deviation is a statistical quantity that measures how spread out the data in a sample are. The symbol for standard deviation is the same as that which is used for surface tension. It is important to pay attention to the data so that values are not mistaken for something else. _________________

Standard Deviation: σ = √ 271.23 kg/m3 / (3-1) = 11.65 kg/m3

Because there is a small discrepancy among the calculated results for the density of Suave Ocean Breeze shampoo, one may employ Chauvenent’s rule to exclude one or more of the date points from the sample. Table 5 contains the values required to employ Chauvenent’s rule.

Table 5: Calculated values required for Chauvenent’s Rule.

Chauvenent’s Rule for Three Data Points C = 1.38

Method # Calculated Density (ρ) (ρ-ρavg)2 1 1032 kg/m3 176.0 kg2/m6

ρavg 1019 kg/m3

2 1010 kg/m3 72.8 kg2/m6 3 1014 kg/m3 22.4 kg2/m6 σ 11.65 kg/m3

Σ = 3056 kg/m3 271.2 kg2/m6

Cσ 16.07 kg/m3

Chauvenent’s Rule: (ρave – Cσ) ≤ ρave ≤ (ρave + Cσ)

Chauvenent’s Rule Calculated for Density: 1003 kg/m3 ≤ ρave ≤ 1035 kg/m3

By the above calculation for Chauvenent’s Rule, none of the data points from the sample are outside of the defined range and therefore cannot be excluded.

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Appendix

Data Usage Sample Calculation of the Density of Ocean Breeze Shampoo (Method Two)

Density of Fluid (ρ): 10.102 g / 10 cm3 = 1.0102 g/cm3 = 1010 kg/m3 Sample Calculation for the Average Surface Tension of Ocean Breeze Shampoo

σ (avg) = [(28.9 + 32.2 + 24.7 + 30.8) dyne/cm] / 4 = 29.1 dyne/cm

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Bibliography

General Chemistry, Fourth Edition J.W. Hill, Ralph H. Petrucci, et al. (2005)

Introduction to Fluid Mechanics, 3rd Edition

William S. Janna (1993)

Fundamentals of Material Science and Engineering: An Integrated Approach W.D. Callister, Jr and D.G. Rethwish (2008)

Using a Hydrometer

http://www.lumcon.edu/education/studentdatabase/hydrometer.asp The Louisiana Universities Marine Consortium

A Manual for the Mechanics of Fluid Laboratory

William S. Janna (2008)