In the Classroom
The Floating Needle F. E. Condon*† Department of Chemistry, The City College of The CUNY, New York, NY 10031 F. E. Condon Jr. Palm Beach County Water Utilities, Florida
It is common knowledge that a steel needle can be floated on water, though it will sink to the bottom of a vessel if introduced point first, its density being 7–8 times that of water. The floating, of course, is a consequence of surface tension. It is also certain that there is a limit to the diameter of the needle, considered as a cylindrical rod, that will permit floating. One cannot, for example, float an eight-penny nail, head and point cut off.
Energy, E +E
descent through air
−D
D
2D
0
surface of water
Depth of immersion, d
descent through water
floating position
BAB′ = 2θ × 2πr = πθr 360 90
C
D' θ
(r - d)
r
s B
d
B'
Surface of water
A
Figure 2. End view of cylindrical rod with radius, r, and length, L, immersed to depth, d. †
334
BB′ = 2s = 2 2dr – d 2
−E
Figure 1. A priori conception of the relationship between the energy of the “floating needle” system as the needle descends from a height above water equal to its diameter to a depth of immersion twice its diameter, D. Scales are arbitrary.
D
A popular view is that surface tension produces a film or membrane on the surface of the water that supports the needle. This view is widely discredited in the literature (1; 2, pp 2, 83–84). Here we have examined the phenomenon in terms of the energy of the system as the needle (or cylindrical rod) is lowered from a height above the surface to complete submersion and beyond. Factors considered are gravitational potential energy, buoyancy, surface energy (the product of surface tension and surface area), and metal–water interfacial energy (the product of interfacial tension and interfacial area). A priori, we considered the floating as a metastable state resulting from a minimum in the curve relating the energy to the distance of descent, as shown in Figure 1. We show here that such an energy minimum arises naturally out of the mathematics needed to evaluate the factors mentioned above. In Figure 2 is presented an end-on view of a cylindrical rod with diameter, D, and radius, r, immersed to a depth, d. The length, L (not shown), is considered to be so long that effects arising from coverage of the ends of the rod can be neglected. From the figure: s 2 = r 2 – (r – d )2 = 2dr – d 2 θ = cos᎑1(r – d )/r = “the angle whose cosine is (r – d )/r”
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90 – θ πr 2θ Area ABB′A = 1 πr 2 – 2 πr 2 – s r – d = –s r –d 180 2 360 Table 1. Values as a Function of Rod Radius, r, and Depth of Immersion, d d/cm
θ/deg
BB′(= 2s)/ cm
BAB′/ cm
Area ABB′A/ cm2
0.1r
25.84
0.872r
0.902r
0.059r 2
0.2r
36.87
1.200r
1.287r
0.164r 2
0.3r
45.57
1.428r
1.591r
0.295r 2
0.4r
53.13
1.600r
1.855r
0.447r 2
0.5r
60.00
1.732r
2.094r
0.614r 2
0.6r
66.42
1.834r
2.318r
0.792r 2
0.7r
72.54
1.908r
2.532r
0.980r 2
0.8r
78.46
1.960r
2.739r
1.173r 2
0.9r
84.26
1.990r
2.941r
1.371r 2
1.0r
90.00
2.000r
3.142r
1.571r 2
1.1r
95.74
1.990r
2.343r
1.771r 2
⯗ 2.0r
⯗ 180.00
⯗
⯗
⯗
0.000r
6.283r
3.142r 2
Journal of Chemical Education • Vol. 78 No. 3 March 2001 • JChemEd.chem.wisc.edu
In the Classroom
(This is needed to calculate the effect of buoyTable 2. Estimated Buoyancy Effect of Water on Gravitational ancy. When multiplied by the length, L, of the Potential Energy Loss of Steel Cylindrical Rod, 0.1 ⴛ 10 cm, As It Falls through Water rod, it provides the volume and hence the mass Cumuof the water displaced as the rod descends Vol. Water Effective Eg loss lative Eg Area ABB ′Aa / Mean through water.) In Table 1 are presented nuD i s p l a c e d / m g ∆d/ d/cm m / g m a s s o f w cm2 m′ b/g loss/ cm3 ergs rod, m′/g merical values of these entities for various depths ergs of immersion, expressed as a function of the 0.000 0.0000000 0.000000 0.000000 0.6087c — 0.000 0.000 rod’s radius, r. 0.005 0.0001475 0.001475 0.001475 0.6072 0.6080 2.979 2.979 We were able to float a standard-sized pa0.010 0.0004100 0.004100 0.004100 0.6046 0.6059 2.969 5.948 per clip, slightly unbent so that no parts were 0.015 0.0007375 0.007375 0.007375 0.6013 0.6030 2.955 8.903 touching each other, diameter about 0.1 cm and 0.020 0.0011175 0.011175 0.011175 0.5975 0.5994 2.937 11.84 length about 10 cm (more nearly 0.093 × 10.5 ⯗ ⯗ ⯗ ⯗ ⯗ ⯗ ⯗ ⯗ cm), but were unable to float a “giant”-sized 0.095 0.007708 0.07708 0.07708 0.5316 — 2.611 53.201 paper clip, diameter about 0.15 cm (more nearly 0.123 cm). Therefore calculations were done 0.100 0.007855 0.07855 0.07855 0.5301 0.5309 2.601 55.802 d for cylindrical steel rods of these dimensions, a See Table 1, column 5. with expectation of finding a minimum in the bAverage of effective mass at indicated distance and the preceding value of effective energy vs distance-of-descent curve for the 0.1- mass. cA density of 7.75 g cm᎑3 was used. cm rod but not for the 0.15-cm rod. d It can be verified that a rod of these dimensions would lose 59.651 ergs falling Calculations of the buoyancy effect are prethough air and 51.954 ergs falling through water. sented in Table 2 for a steel rod, r = 0.05 cm. As the rod descends through water, its effective mass decreases continuously by an amount equal to the mass Table 3. Energy Changes for 0.1 ⴛ 10 cm Cylindrical of the water displaced. Following is a list of symbols and abSteel Rod Falling through Water breviations used. Surface dyne: Unit of force, 1 g cm s᎑2. One dyne equals 10᎑2 mN. Note that 1 dyne cm᎑1 and 1 mN m᎑1 are equivalent units of surface tension. erg: Unit of energy, 1 g cm2 s᎑2. One erg equals 10᎑7 joule. π: “pi” = 3.1416. g: Acceleration produced by gravity, 980 cm s᎑2. ρ1: Density of water, 1 g cm᎑3. ρ2: Density of metal: 7.75 g cm᎑3 for iron, 2.70 g cm᎑3 for aluminum.1 σ1: Surface tension of water, 72 dyne cm᎑1. 1 σ2: Metal–water interfacial tension, dyne cm᎑1. Zero or assigned values. d: Depth of immersion, cm. D: Diameter of cylindrical rod, cm. r: Radius of rod, cm. m: Mass of rod, g. m′: Mass of rod adjusted for buoyancy, g. mw: Mass of water displaced by rod, g. E: Energy: loss, change, or total, as described, ergs. L: Length of rod, cm.
In Table 2, a stepwise approach was taken to approximate the continuous loss of mass by the rod as it descends through water. An effective mass was calculated after each 0.005 cm of descent. The average of this with its previous value gave a “mean” effective mass, m′ (column 6). Multiplication by the acceleration of gravity and the interval of descent (0.005 cm) gave an estimate of the gravitational potential energy loss for that interval (column 7). Summation of these gave the total energy loss for each depth of immersion. It is postulated that, as the rod descends through water, aqueous surface energy of the covered surface, BB′ × L, is lost and is replaced by metal-water interfacial energy of the wetted surface, BAB′ × L. In Table 3 are presented estimates of the
Total E Energy Eg loss b / loss for Loss, Es / ergs σ2 = 0 ergs
d/cm
BB′ (= 2s)/ cma
BB′ × L/ cm2
0.000
0.0000
0.000
0.0
0.00
0.005
0.0436
0.436
31.4c
2.98
34.4
0.010
0.0600
0.600
43.2
5.95
49.2
0.015
0.0714
0.714
51.4
8.90
60.3
⯗
⯗
⯗
0.050
0.1000
1.000
72.0
29.01
101.0
0.055
0.0995
0.995
71.6
31.79
103.4
0.060
0.0980
0.980
70.6
34.54
105.1
0.065
0.0954
0.954
68.7
37.27
106.0
0.070
0.0917
0.917
66.0
39.97
106.0
0.075
0.0866
0.866
62.4
42.66
105.1
0.080
0.0800
0.800
57.6
45.32
102.9
0.085
0.0714
0.714
51.4
47.96
99.4
0.090
0.0600
0.600
43.2
50.59
93.8
0.095
0.0436
0.436
31.4
53.20
84.6
0.100
0.0000
0.000
0.0
55.80
55.8
⯗
⯗
0.00
⯗
a See
Table 1, column 3. b From Table 2. cA surface tension of water of 72 dyne cm᎑1 was used.
surface energy loss for various depths of immersion differing by 0.005 cm. To these have been added the corresponding losses of gravitational potential energy from Table 2. These totals (column 6), which would correspond with a metal– water interfacial tension of zero, are presented graphically in Figure 3. An energy minimum is found at a depth of immersion about 7⁄10 the diameter of the rod. In the absence of a value for the metal–water interfacial tension, we have used “zero”, as in Table 3 and Figure 3, or have
JChemEd.chem.wisc.edu • Vol. 78 No. 3 March 2001 • Journal of Chemical Education
335
In the Classroom Table 4. Estimation of Interfacial Tension Energy and Net Energy Loss for Cylindrical Steel Rod, 0.1 ⴛ 10 cm, Falling through Water d/cm
BAB′ a/ cm
BAB′ × L/ cm2
0.000
0.0000
0.000
0.005
0.0451
0.010
0.0644
Ei (gain)/ ergs b
100
᎑(Eg + Es) c/ ᎑Enet / ergs ergs
0.00
0.0
0.0
0.451
8.01
34.4
26.4
0.644
11.44
49.2
37.8
⯗
E / ergs
⯗
descent through air -0.1
-0.05
0.05
0.10
0.15
0
⯗
0.2
d / cm
⯗
⯗
⯗
0.050
0.1571
1.571
27.90
101.0
73.1
σ2 = 17.76 dyne cm᎑1
0.055
0.1671
1.671
29.68
103.4
73.7
0.060
0.1773
1.773
31.49
105.1
73.6
0.065
0.1876
1.876
33.32
106.0
73.0
descent through water
0.070
0.1983
1.983
35.22
106.0
72.8
0.075
0.2095
2.095
37.21
105.1
67.9
0.080
0.2215
2.215
39.34
102.9
63.6
0.085
0.2347
2.347
41.68
99.4
57.7
0.090
0.2498
2.498
44.36
93.8
49.4
0.095
0.2691
2.691
47.79
84.6
36.8
0.100
0.3142
3.142
55.80
55.8
0.0
σ2 = 0
-100
Figure 3. Calculated relationships between the energy, E, and the depth of immersion, d, of a steel cylindrical rod 0.1 × 10 cm, as it descends from a height above water equal to its diameter to a depth of immersion twice its diameter.
a See
Table 1, column 4. interfacial tension, σ 2, of 17.76 dyne cm᎑1 was used. c From Table 3, column 6.
E / ergs
bAn
used assigned values (“Harkins obtained the value 375 dynes/ cm for the interfacial tension of water against mercury”; 2, p 51). In the case of the 0.1 × 10-cm steel rod, a value for the interfacial tension of 17.76 dyne cm᎑1 is of particular interest, as it leads to an energy minimum at a depth of immersion about 0.55 times the diameter of the rod and to a net energy of zero at the point of complete immersion. The data, presented in Table 4 and plotted in Figure 3, are intended to be illustrative and not definitive. Similar calculations were made for a steel rod, 0.15 × 10 cm. The data are not given here because they can be reproduced by the reader. Results are presented graphically in Figure 4. An energy minimum is found at a depth of immersion of about 0.12 cm (80% of the diameter). According to this theory, the rod (the “giant”-sized paper clip with a diameter of 0.123 cm) should float at that depth, but of course it did not. We theorized that at a depth so near complete submersion, capillarity may come into play, causing water to cover the top exposed surface of the rod and “sinking” it. We tried coating the top surface of the rod (a giant paper clip about 15.5 cm long) with petroleum jelly to repel wetting, but this subterfuge was not effective. When the paper clip was coated on the top side with candle wax, however, it floated— for several hours, in one successful experiment! (About 3 g of candle wax, density about 0.9 g cm᎑3, was melted in a flatbottomed Teflon-coated frying pan. The paper clip was placed in a pool of the molten wax not deep enough to cover it. The clip was removed with tweezers and placed, coated side up, on paper towel. When cool, the clip was floated on water, coated side up. It could not be floated coated side down.) Use of a silicone sealant, spread by finger on the top of a giant paper clip, was also effective. Such experiments were repeated successfully several times. The result supports the
336
200
descent through air 100
-0.1
0
0.1
0.2
0.3
d / cm
-100
-200
descent through water
-300
Figure 4. Calculated relationship between the energy, E, and the depth of immersion, d, of a cylindrical steel rod, 0.15 × 10 cm, as it descends from a height above water equal to its diameter to a depth of immersion twice its diameter, assuming a metal–water interfacial tension of zero.
theory that capillarity may be the reason the uncoated paper clip could not be floated. One would expect to be able to float rods of larger diameter if made from less dense metal, such as aluminum (density 2.70 g cm᎑3, a little more than a third that of iron or steel1). We were able to float an aluminum nail, head and point cut off, with a diameter of 0.218 cm.2 Calculations were made for an aluminum rod, 0.3 × 10 cm. With a metal– water interfacial tension of zero, an energy minimum was
Journal of Chemical Education • Vol. 78 No. 3 March 2001 • JChemEd.chem.wisc.edu
In the Classroom
unrealistic, in view of our difficulty in floating a rod with a diameter of 0.123 cm. A wetting scenario is favored, therefore. In conclusion, and in the absence of definitive values for metal–water interfacial tension:
E / ergs descent through air
-0.3
500
0.3
0
0.6
d / cm
-500
descent through water
-800
Figure 5. Calculated relationship between the energy, E, and the depth of immersion, d, of a cylindrical aluminum rod, 0.3 × 10 cm, as it descends from a height above water equal to its diameter to a depth of immersion twice its diameter, assuming a metal–water interfacial tension of zero.
found at a depth of immersion about 0.255 cm, 85% the diameter of the rod, as shown in Figure 5. Two other aspects should be considered: “contact angle” and “wetting” vis-à-vis “nonwetting”. A contact angle different from that implied in Figure 2 would be expected to promote floating, as it would require extension of the liquid surface (3). (Imagine the rod in a depression with sloping sides, for example.) Such extension of the liquid surface is undoubtedly an important factor aiding the flotation of razor blades and other pieces of metal (e.g., see 2, p 84 and Fig. 21). If by “wetting” it is meant that the molecules of liquid and solid are close enough together for intermolecular forces of attraction to be operative, interfacial tension would be expected to be low, 0–20 dyne cm᎑1, say, as in our foregoing treatment. If by “nonwetting” it is meant that there is a thin layer of vapor-saturated air between the liquid and solid surfaces, the interfacial tension, it seems to us, would be essentially the surface tension of the liquid, 72 dyne cm᎑1 in the case of water. Calculations were done using this value of interfacial tension for a steel rod, diameter 0.2 cm. An energy minimum was found at 70% submersion, which seems
1. The floating of a cylindrical metal rod on water is essentially a consequence of recovery of aqueous surface and surface energy, which commences when the rod has submerged to 50% of its diameter. 2. Positive values for metal–water interfacial tension aid flotation by adding energy to the system, reducing the depth of immersion at which floating takes place. 3. The limiting diameter for rods that can be floated is about 0.125 cm for iron and 0.3 cm for aluminum. 4. Capillarity, by causing water to creep over the top of a partially submerged rod and thus to sink it, helps to place an upper limit on the diameter for rods that can be floated, as it was found that— 5. —coating the top side of a steel rod with certain waterrepellant substances made it possible to float a rod that otherwise would not float.
Notes 1. Physical constants are “nominal”. Values, not always in agreement, may be found in handbooks, textbooks, unabridged dictionaries, and encyclopedias. One may read for iron or steel, “a metal with a density about 7.75 g cm᎑3”; for aluminum, “a metal with a density about 2.70 g cm᎑3”; and for water, “a liquid with a density about one g cm᎑3 and surface tension about 72 dyne cm᎑1”. 2. Four standard- and four giant-sized steel paper clips and four pieces of aluminum nails (from various sources, but similar to and including some of those floated) were weighed on a Mettler balance. Their diameters were measured with a micrometer and their lengths, with a millimeter rule. From these data, densities were calculated. Results for the steel paper clips ranged from 7.31 to 8.43 g cm᎑3 and averaged 7.87 g cm᎑3. For the aluminum nails, results ranged from 2.55 to 2.65 g cm᎑3 and averaged 2.61 g cm᎑3. We elected to use the values given above,1 however, in our calculations.
Literature Cited 1. Adam, N. K. The Physics and Chemistry of Surfaces; Dover: New York, 1968; Preface and pp 1–6. 2. Burdon, R. S. Surface Tension and the Spreading of Liquids; University Press: Cambridge, UK, 1949. 3. Leading references may be found in Kabza, K.; Gestwicki, J. E.; McGrath, J. L. J. Chem. Educ. 2000, 77, 63–65.
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