Calculating surface tension with the Capillary Equation

[Allen Montgomery]

Homework Statement

Hello everyone, I am doing an experiment and I've hit a snag with my calculations. I am looking at how concentrations of soap affect surface tension in water. I am have been using the capillary equation and capillary tubes for my calculations. I have practiced this method for 100% water and I have gotten the value of 2.36E-05 as my surface tension of water at 70 degrees F. I would assume that doing this correctly would give me the surface tension for water at 70 degrees F but as I look at other sources the values appear to be .0644 N/M. I am unsure and am looking for help as to where my methodological errors would occur for my value to be this off. I feel like I am missing something but I am unsure as to what it is. (All units should be SI)

Homework Equations

Here is the capillary equation and the variables:

s=(phga)/2

s= surface tension
h= sifference in height between depth of water and the liquid in the capillary tube .0193 m
p= density (of water) =1
g= gravity (9.8) m/s
a= raidus (2.5x10-4) m

The Attempt at a Solution

Raw data (h values)

1.95cm
1.95cm
1.9cm

Av: (1.95+1.95+1.9)/3 = 1.93cm = .0193m = h

h= .0193
p =1
g= 9.8 m/s
a= 2.5x10-4 m

s=((1)(9.8)(2.5x10-4)(.0193))/2

s= 4.7285x10-5
s= 2.364E-5 (final answer)

 

[Chestermiller]

What are the units of rho?

 

[Allen Montgomery]

What are the units of rho?

I am unsure as to what you are referring to, do you mean the radius of the hole? In that case it is m

 

[Chestermiller]

I am unsure as to what you are referring to, do you mean the radius of the hole? In that case it is m

The density

 

[Allen Montgomery]

Should be g/ml

 

[Chestermiller]

You’re using metric units, right? Kg/m^3?

 

[Allen Montgomery]

I am supposed to be using SI units, did I miss a conversion?

 

[Chestermiller]

I am supposed to be using SI units, did I miss a conversion?

What about g?

 

[Allen Montgomery]

I will have to take a look at that in my calculations, I clearly missed that. I'll post my new results when I get a chance.

Thank you for the help, sir!

 

[haruspex]

I am supposed to be using SI units, did I miss a conversion?

g/ml (grams per millilitre) is not SI. (At least, not since about 1960.)
As Chet suggested, SI would be kg/m³. There's a substantial ratio between the two.
But with that correction, you will still have less than half the value you found elsewhere. And by the way, I think the value you quote is for 70C, not 70F.

 

[Allen Montgomery]

And by the way, I think the value you quote is for 70C, not 70F.[/QUOTE]

Thank you haruspex, you are correct the reference site I used was in F, after some conversion my value should be around .073-.074 N/m range.

I did make the necessary conversion for my density which did solve my issue concerning the "x10^-4" and put my value in the correct range.

However, the new value I have gotten for water is .02359 kg/s^2 which is still off. I am unsure as to where my error has occurred.
g = 9.8m/s^2
h = .0193m
a = 2.5x10^-4
p = 998.2 (more precise value for water than 1)

{(9.8)( .0193)(2.5x10^-4)(998.2)}/2 = .02395 kg/s

Also, Is kg/s^2 equivalent to N/m or is there a conversion I've missed again?

Thank you all for the help so far!

 

[Chestermiller]

And by the way, I think the value you quote is for 70C, not 70F.

Thank you haruspex, you are correct the reference site I used was in F, after some conversion my value should be around .073-.074 N/m range.

I did make the necessary conversion for my density which did solve my issue concerning the "x10^-4" and put my value in the correct range.

However, the new value I have gotten for water is .02359 kg/s which is still off. I am unsure as to where my error has occurred.
g = 9.8m/s
h = .0193m
a = 2.5x10^-4
p = 998.2 (more precise value for water than 1)

{(9.8)( .0193)(2.5x10^-4)(998.2)}/2 = .02395 kg/s

Also, Is kg/s equivalent to N/m or is there a conversion I've missed again?

Thank you all for the help so far!

The units of surface tension are kg/s^2, which is the same as N/m.

 

[Allen Montgomery]

Ok, that great, and I did accidentally write m/s when it was supposed to be m/s^2.

Any idea on what could have caused my value (.02359 kg/m^s) to be so far off from .073-0.74 N/m that it should be? That is the biggest issue I am having to deal with.

(Also estimated the value for Surface tension of water at 70 degrees F from the second chart in this source https://www.engineeringtoolbox.com/water-surface-tension-d_597.html)

 

[haruspex]

Ok, that great, and I did accidentally write m/s when it was supposed to be m/s^2.

Any idea on what could have caused my value (.02359 kg/m^s) to be so far off from .073-0.74 N/m that it should be? That is the biggest issue I am having to deal with.

(Also estimated the value for Surface tension of water at 70 degrees F from the second chart in this source https://www.engineeringtoolbox.com/water-surface-tension-d_597.html)

Was the tube really only 0.5mm internal dismeter?
Or maybe some detergent residue on the glassware?

 

[Allen Montgomery]

Was the tube really only 0.5mm internal dismeter?
Or maybe some detergent residue on the glassware?

I used a new capillary tube for every trial so that would eliminate the detergent residue (it was a variable I discussed in my paper).

On the jar of tubes, it read 0.9mm-0.5mm, which I assumed that .5mm referred to the hole and .9mm as the total diameter. Could that be an issue?
I will double check the jar when I have a chance tomorrow though.

 

[haruspex]

I used a new capillary tube for every trial so that would eliminate the detergent residue (it was a variable I discussed in my paper).

On the jar of tubes, it read 0.9mm-0.5mm, which I assumed that .5mm referred to the hole and .9mm as the total diameter. Could that be an issue?
I will double check the jar when I have a chance tomorrow though.

It's not just the tube. Any receptacle the water contacted would need to be clean of surfactants.
If the tubes all had an outside diameter of about 1mm then your interpretation is probably right.

 

[Allen Montgomery]

I see what you mean, I used new plastic cups for each trial but I will have to evaluate my experiment in the near future.

But besides that are there any mathematical errors that could be of an issue?

 

[haruspex]

are there any mathematical errors that could be of an issue?

Not that I can see.

 

[Chestermiller]

The equation you used assumes that the contact angle was 90 degrees. Was it?

 

[haruspex]

The equation you used assumes that the contact angle was 90 degrees. Was it?

Don't you mean zero degrees?
Might be hard to judge without a microscope in a half mm tube!
As I recall, water on glass should have a zero contact angle.

 

[Chestermiller]

Don't you mean zero degrees?
Might be hard to judge without a microscope in a half mm tube!
As I recall, water on glass should have a zero contact angle.

It depends on which angle is being referred to. I have seen it both ways. But, by your definition, it could be as much as 20 degrees on glass. Still, this is not enough to explain the OPs experimental difference. Maybe the height was in inches instead of cm?

 

[Allen Montgomery]

Don't you mean zero degrees?
Might be hard to judge without a microscope in a half mm tube!
As I recall, water on glass should have a zero contact angle.

I can certainly say that I tried to hold the tube as upright in the experiment but I may have not been that precise with the angle of contact.

 

[haruspex]

Maybe the height was in inches instead of cm?

That would certainly get close.

 

[Allen Montgomery]

It depends on which angle is being referred to. I have seen it both ways. But, by your definition, it could be as much as 70 degrees on glass. Still, this is not enough to explain the OPs experimental difference. Maybe the height was in inches instead of cm?

I did check but the height was always measured in cm for my raw data then converted to m later on. I will take another look.

 

[haruspex]

may have not been that precise with the angle of contact.

How would you have known? Was there a microscope in the set up?

 

[Allen Montgomery]

How would you have known? Was there a microscope in the set up?

No there was not, I simply held the tube with my hand and marked the depth and total height with a marker.

 

[haruspex]

No there was not, I simply held the tube with my hand and marked the depth and total height with a marker.

So you would not have been able to judge the contact angle. But I doubt that was a problem.
Holding the tube in your hand sounds like more of a risk. You really should have had it clamped.
Where were you making the marks? On the cup?

 

[Allen Montgomery]

So you would not have been able to judge the contact angle. But I doubt that was a problem.
Holding the tube in your hand sounds like more of a risk. You really should have had it clamped.
Where were you making the marks? On the cup?

No there were two marks on the capillary tube itself, one for the depth (roughly 2.7cm) and one for total height (4.6, cm on average). The difference between these values should be the h values in the capillary equation.

 

[haruspex]

No there were two marks on the capillary tube itself, one for the depth (roughly 2.7cm) and one for total height (4.6, cm on average). The difference between these values should be the h values in the capillary equation.

So you were making these marks by hand, at some depth within the surrounding cup maybe?, holding the tube with the other hand, while looking down into the cup from above... All sounds a bit awkward.
Presumably the base of the tube was resting on the base of the cup. I don't think that should affect the result, but it does not sound ideal.

 

[Allen Montgomery]

So you were making these marks by hand, at some depth within the surrounding cup maybe?, holding the tube with the other hand, while looking down into the cup from above... All sounds a bit awkward.
Presumably the base of the tube was resting on the base of the cup. I don't think that should affect the result, but it does not sound ideal.

I was making these markings by hand, although I used a short cup so I could look at it and mark it from the side. But yes it was a bit awkward but I did get mostly consistent results in the experiment

The base of the tube was resting on the base of the cup for every trial as well.