Month: July 2012

Exponential Growth/Decay Models (Summary)

A quick summary (more details here):

The equation that describes exponential growth is:
Exponential Growth: N = N_0 e^{rt}

where:

  • N = number of cells (or concentration of biomass);
  • N0 = the starting number of cells;
  • r = the rate constant, which determines how fast growth occurs; and
  • t = time.

You can set the r value, but that’s a bit abstract so often these models will use the doubling time – the time it takes for the population (the number of cells, or whatever, to double). The doubling time (td) can be calculated from the equation above by:

t_d = \frac{\ln 2}{r}

or if you know the doubling time you can find r using:

r = \frac{\ln 2}{t_d}

Finally, note that the only difference between a growth model and a decay model is the sign on the exponent:

Exponential Decay: N = N_0 e^{-rt}

Decay models have a half-life — the time it takes for half the population to die or change into something else.

Live Bears

Watch brown bears catch salmon in Katmai National Park on a live webcam:

Every year over a hundred Brown Bears descend on a mile long stretch of Brooks River to feast on the largest Sockeye Salmon run in the world.

— bears: brown bear & salmon cam – brooks falls on explore.org

Live video from Brooks Falls, Alaska.

Switch: Converting Audio Files for Sound Effects

Adding sound to webpages is pretty easy with HTML5’s Audio tag (as Jean-Baptiste Jung demonstrates), but since different browsers can only handle certain, different types of sound files, you’ll often need to convert your sound into .wav, .ogg, and .mp3 formats so they can work for everyone.

Switch Sound File Converter does just what it says, and is free for non-commercial use.

Anatronica: 3d Anatomy Online

Screen captures from Anatronica's Anatomy 3D Systems website. The digestive system is highlighted, while the skeletal system is shown semi-transparently for context.

Anatronica has an excellent, online, 3d viewer for the anatomy of the human torso. While it’s not quite the same as a physical model, it’s pretty good as a study guide for middle schoolers.

Endothermic Reactions: Vinegar and Baking Soda

A quick and simple experiment that demonstrates endothermic reaction and can include a discussion of ionic and covalent bonds. Mixing baking soda and vinegar together drops the temperature of the liquid by about 4 °C in one minute. (Note that while the temperature drops and the reaction looks endothermic, it’s actually not — other things cause the cooling. However, since it looks like an endothermic reaction I use it as a first approximation of one.)

Ingredients

  • 3 g baking soda – (sodium bicarbonate – NaHCO3)
  • 60 ml vinegar – (acetic acid – CH3COOH)
  • 200 ml styrofoam cup (needs to be big enough to contain the bubbles).
  • thermometer

Procedure

Add the baking soda to the vinegar in the styrofoam cup. Measure the temperature while stirring for about a minute.

Results

Time (t) Temperature (°C)
0 25
15 24
30 21
60 21

Discussion

The chemical reaction between baking soda (sodium bicarbonate) and vinegar (acetic acid) can be written:

NaHCO3 + CH3COOH —-> CO2 + H2O + CH3OONa

The products of the reaction are carbon dioxide gas (which gives the bubbles), water, and sodium acetate.

However, a more detailed look shows that for the reaction to work the two chemicals need to be dissolved in water. Dissolving these ionic compounds causes the two ions to separate. Dissolved baking soda dissociates into a sodium and a bicarbonate ion:

sodium bicarbonate —-> sodium ion + bicarbonate ion

NaHCO3 —-> Na+ + HCO3–

Why doesn’t the bicarbonate break into smaller pieces? Because it’s atoms are bonded together more tightly by covalent bonds.

Similarly, the acetic acid in vinegar dissociates into:

acetic acid —-> hydrogen ion + acetate

CH3COOH —-> H+ + CH3COO–

This video has a nice overview of ionic versus covalent bonding.

References

More detail about the reaction can be found at: