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Homework Help: Surface Area to Volume Ratio

  1. Sep 28, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose a lipid droplet is essentially a sphere with a diameter of 1 cm.

    For A – D below, the area of a sphere is 4π r2, and volume is 4/3πr3.

    For π, use a value of 3.14.

    Show your calculations neatly in the space provided to receive credit for your answer.

    Round all answers to the nearest tenth.


    B. Now, suppose that this sphere were emulsified into 100 essentially equal-sized droplets. What is the average surface area to volume ratio of each droplet?


    C. How much greater is the total surface area of these 100 droplets compared to the original single droplet?

    D. How much did the total volume change as a result of emulsification?

    2. Relevant equations

    Surface Area / Volume

    3. The attempt at a solution

    What is the surface area to volume ratio of the droplet?

    Answer: 6.0

    Surface Area / Volume = 4 x (3.14) x .5^2 / (4/3) x 3.14 x .5^3 = 6.04 or 6.0


    I figured out A but am stuck on part B through D. Can you help? Thank you! Thank you!

    I don't know how the equation evolves when the sphere is emulsified into 100 equal droplets. I know that the surface area
    must increase but I cannot get it to work :(
     
  2. jcsd
  3. Sep 28, 2010 #2
    When the original droplet splits into 100 equal size droplets, each one of these smaller droplets has an volume that is 1/100 of the original droplet. Use this and the equation for the volume of a sphere to calculate the radius of the smaller droplet, and then knowing their radius you can calculate their surface area.
     
  4. Sep 28, 2010 #3

    LCKurtz

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    You have already posted this question in the Calculus forum. Don't multi-post.
     
  5. Sep 28, 2010 #4

    eumyang

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    ... and in the Topology & Geometry forum...
     
  6. Sep 28, 2010 #5
    Sorry im stupid when it comes to math so i posted at three places since i really didnt know where to put :-(
     
  7. Sep 28, 2010 #6
    I don't understand. I thought that radius is already understood given that we were given the diameter of 1 cm. I was under the impression that radius is .5 cm / 100 droplets which equals 0.005 cm. In other words, can I plug in "0.005" whenever "r" in the original surface area / volume equation? Maybe, I'm I doing something wrong?
     
  8. Sep 29, 2010 #7
    No, you cannot just divide the radius by 100. If you break a sphere into two and make two smaller sphere's from the halves their radius will not be halve of the radius the original sphere. However their volume will be halve the volume.
     
  9. Sep 29, 2010 #8

    D H

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    Does breaking up the droplet into 100 smaller droplets change the mass or the density one iota? What does this mean with respect to question D? What does that answer mean with respect to questions B and C?

    BTW, what was question A?
     
  10. Sep 29, 2010 #9

    berkeman

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    (Multiple threads merged. We trust this won't happen again.)
     
  11. Sep 29, 2010 #10
    Question A was What is the surface area to volume ratio of the droplet?

    Answer: 6.0

    Surface Area / Volume = 4 x (3.14) x .5^2 / (4/3) x 3.14 x .5^3 = 6.04 or 6.0
     
  12. Sep 29, 2010 #11
    I'm sorry for the trouble again.
     
    Last edited: Sep 29, 2010
  13. Sep 29, 2010 #12
    Thanks for the clarification. I will find the volume right now and will post the latter part of my answer in a few hours, after my class.


    Volume = 4/3πr3

    Volume = (4/3) x 3.14 x 0.005^3 = 5.23333333334e-7


    Or in the alternative:

    It could be



    Volume = 4/3πr3


    Volume = (4/3) x 3.14 x 0.5^3 = .5233333333; then divided by 100 = .00523333333333


    I wonder which among the two is correct?
     
    Last edited: Sep 29, 2010
  14. Sep 29, 2010 #13
    Second answer is correct. The questions is about relation between lengths, surface area and volume. Think about how these relate. Try a simpler shape than the sphere to get some feel for it first perhaps.

    Take a cube for example:
    a cube with sides of length 2 has volume 2x2x2=8

    Cut it in half so that you get two cubes of each 2x2x1, so each has volume 4.

    This not the same as just halving the lengths of the sides and then calculating
    1x1x1=1.
     
  15. Sep 29, 2010 #14
    Gerben... I worked out the equation based on your thoughts and some research. I hope this looks right or kinda right. Can you take a look B, C, and D to see if its interpreted correctly? Thank you so much. You are a life saver!


    .00523333333333 = (4/3) x 3.14 x r^3
    Multiply both sides by 3 to get rid of the 3 from (4/3)
    .00523333333333 x (3) = 4 x 3.14 x r^3
    .0157 = 4 x 3.14 x r^3
    Divide (4 x 3.14) against .0157 to get rid of it in the right side
    .00125 = r^3
    Cube it!
    ³√x.00125 = radius = .108
    Substitute .108 in the Surface Area Formula SA= 4 x (3.14) x (r ^2)
    4 x (3.14) x (.108^2)
    12.56 x (0.011664) =0 .1465 = Surface Area
    Plug in the numbers in the equation Surface Area / Volume
    0.1465 / .00523333333333 = 27.99
    B. Now, suppose that this sphere were emulsified into 100 essentially equal-sized droplets. What is the average surface area to volume ratio of each droplet?
    27.99 = (28.0)
    C. How much greater is the total surface area of these 100 droplets compared to the original single droplet? 466.7 Greater
    28.0 x 100 = 2800
    2800/ 6 = 466.7
    D. How much did the total volume change as a result of emulsification? Total volume of 100 droplet is the same as the BIG droplet ( before being emulsified)
     
  16. Sep 29, 2010 #15
    Also in case you are wondering- I got the 6.0 from my answer to question "A"

    What is the surface area to volume ratio of the droplet?

    Answer: 6.0

    Surface Area / Volume = 4 x (3.14) x .5^2 / (4/3) x 3.14 x .5^3 = 6.04 or 6.0
     
  17. Sep 30, 2010 #16
    B and D seem okay. When you say "Cube it" you mean the inverse namely "take cube root". Cube it means take it to the 3rd power.

    You could also first write it out in symbols so you do not have to round off in between like this:

    Volume of large droplet:
    [tex] V = \frac{4}{3}\pi(\frac{1}{2})^3 = \frac{4}{3}\pi\frac{1}{8} [/tex]

    Volume of small droplet:
    [tex] V_s = (\frac{4}{3}\pi\frac{1}{8})/100 = \frac{4}{3}\pi\frac{1}{800} [/tex]

    Get radius of of small droplet from equation:
    [tex] V_s = \frac{4}{3}\pi r_s^3 [/tex]
    [tex] \frac{4}{3}\pi r_s^3 = \frac{4}{3}\pi\frac{1}{800} [/tex]
    [tex] r_s^3 = \frac{1}{800} [/tex]
    [tex] r_s = \sqrt[3]{\frac{1}{800}}[/tex]

    Surface area of small droplet:
    [tex] A_s = 4\pi r_s^2 = 4\pi \left(\sqrt[3]{\frac{1}{800}} \right)^2 \approx 0,1458[/tex]

    Surface area to volume ratio of small droplet:
    [tex] A_s/V_s =\left. 4\pi \left(\sqrt[3]{\frac{1}{800}} \right)^2 \right/
    \frac{4}{3}\pi\frac{1}{800} =
    3\cdot 800 \cdot \left(\sqrt[3]{\frac{1}{800}} \right)^2 \approx 27.8495[/tex]

    For C, you need to calculate: "Surface area all small droplets relative to surface area large droplet"
    but you used "surface area to volume ratio" you should just divide the surface area of all small droplets by the surface area of the large droplet.
     
  18. Sep 30, 2010 #17
    For C, the professor's question was "How much greater is the total surface area of these 100 droplets compared to the original single droplet?"



    Should I subtract instead of dividing? It seems like his question can be interpreted in multiple ways!

    Total Surface Area of 100 Small Droplets = 0.1465 x 100 droplets = 14.65

    14.65 (The Total SA of 100 Droplets) - 3.14 (SA of the Original Surface Area) = 11.48

    Surface area is greater by 11.48.

    Or if I go by your recommendation:

    14.65 (The Total SA of 100 Droplets) / 3.14 (SA of the Original Surface Area) = 4.67 or 4.7. Total Surface Area of the 100 droplets is 4.7 times larger.

    What units can use to express surface area and volume given that the original droplet had a 1 centimeter diameter. Millimeters maybe?

    You are heaven sent gerben! I'm very grateful for taking the time to answer my questions!
     
  19. Sep 30, 2010 #18
    "How much greater" seems to indicate "how many times as large", so I guess he wants you to divide the two.

    Otherwise he should have asked "how much more surface area do the small droplets have than the large droplet".

    Since you used centimeters for the radius (0.5 cm) the surface area is in cm2 and the volume in cm3 .
     
  20. Sep 30, 2010 #19
    Thank you Thank you Gerben!!!:biggrin:
     
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