# physical chemistry

## Chemical Kinetics of Valentine’s Day

If the members of group A and group B want to form a union AB it can be described by the following chemical equation.
$$! \text{A} + \text{B} \rightarrow \text{AB}$$
which will have a rate constant of
$$! R = k[\text{A}][\text{B}]$$
Assuming this is an elementary process we can solve for the rate of this reaction by the introduction of a progress variable $$x$$.
$$! x = ([\text{A}]_0 – [\text{A}]_t) = ([\text{B}]_0 – [\text{B}]_t)$$
Substituting $$\frac{dx}{dt}$$ for $$R$$ yields…
$$! \frac{dx}{dt} = k([\text{A}]_0 – x)([\text{B}]_0 – x)$$
And to determine the time behavior we simply integrate.
$$! \int_{x(0)}^{x(t)} \frac{dx}{([\text{A}]_0 – x)([\text{B}]_0 – x)} = k\int_0^t dt$$
Using the method of partial fractions
$$! \int_0^x \frac{dx}{([\text{A}]_0 – [\text{B}]_0)([\text{B}]_0 – x)} – \int_0^x \frac{dx}{([\text{A}]_0 – [\text{B}]_0)([\text{A}]_0 – x)} = k\int_0^t dt$$
Integrating…
$$! -\frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln\left([\text{B}]_0 – x\right)_0^x + \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln\left([\text{A}]_0 – x\right)_0^x = kt$$
Grouping…
$$! \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln\left(\frac{([\text{A}]_0 – x)}{([\text{B}]_0 – x)}\right)_0^x = kt$$
Evaluating this from 0 to $$x$$
$$! \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln\left(\frac{([\text{A}]_0 – x)}{([\text{B}]_0 – x)}\right) – \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln\left(\frac{([\text{A}]_0 – 0)}{([\text{B}]_0 – 0)}\right) = kt$$
$$! \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln \left(\frac{([\text{A}]_0 – x)}{([\text{B}]_0 – x)}\right) – \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln \left(\frac{[\text{A}]_0}{[\text{B}]_0}\right) = kt$$
Remembering that $$[\text{A}]_0 – x = [\text{A}]_t$$
$$! \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln \left(\frac{[\text{A}]_t}{[\text{B}]_t} \right) – \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln \left(\frac{[\text{A}]_0}{[\text{B}]_0}\right) = kt$$
Simplifying, we finally have an expression for the union of two reactive groups of people on Valentine’s day.
$$! \frac{1}{([\text{A}]_0 – [\text{B}]_0)}\ln \left(\frac{[\text{A}]_t[\text{B}]_0}{[\text{B}]_t[\text{A}]_0} \right) = kt$$

May the rate constant ($$k$$) be large today!

Mitch

By February 14, 2009 7 comments

## Light Powered Motor and Experiment Vlogging

Most of you probably read the last issue of C&EN with the spiffy carrot loving cover story (good for me because I love carrots, but have never tried those ugly-looking BetaSweets). Inside, however, there was an extremely interesting little article in the “Science and Technology Concentrates” about light-driven pulleys turning a plastic motor.

Now photo mobile polymer materials have been around for quite a while, at least from my perspective seeing as how I wasn’t even in highschool when the big Nature paper came out. Some might remember the Nature 1999 Sep 9;401(6749):152-5 Koumura et al. paper titled “Light-Driven monodirectional molecular rotor”. Although back then, the rotation was monodirectional around a C-C double bond in a chiral, helical alkene. It was activated by UV light or a change in temperature and the motor was based on light-induced cis-trans isomerizations that caused 180 degree rotations followed by thermally controlled helicity inversions, which basically nullified half a rotation. Four isomerizations resulted in 1 complete cycle.

Well this was pretty darn cool but we’ve come a long way since then. As expected, and as Koumura said, structurally modified chiral alkenes played the central role in the development of these molecular motors that were beginning to interest the MEMS people (MEMS stands for Micro-Electromechanical Systems…I am pretty sure).

In ter Wiel MK et al. introduced the worlds smallest artificial light-driven motor using 28 carbon atoms and 24 hydrogen atoms.

Reprinted with permission from American Chemical Society: Journal of the American Chemical Society (Nov. 2003).  It also had a dramatic speed increase over the original designs, at a whopping 18s half-life at the fastest step. Even if it wasn’t going to be turning any relevant loads any time soon, it was a dramatic improvement over the original concept 4 years earlier. Still, even though some clever O-chem tricks made the motor better, it still operated on the same 4-step cycle that Koumura’s did back in 99′. Even recently, in Org. Biomol. Chem., 2008, 6, 507 – 512, DOI: 10.1039/b715652a, Pollard et al. report on substituting naphthalene moieties for phenyl moieties, in order to better control the speed of the motors, and to enable the design and synthesis of more complex systems.

Meanwhile, the MEMS people came up with interesting designs similar to this: “A five micron wide resin structure, with a shape resembling a lawn sprinkler, rotates when illuminated by a laser beam. Tiny rotors like this one may someday power micromechanical systems (MEMS), or twist molecules to measure their mechanical properties.” Reported by: Péter Galajda; Pál Ormos, Applied Physics Letters, 8 January, 2001.

There was quite a bit of work done focusing on creating rotors that responded to laser light, although the practical applications of such devices aren’t as numerous as the devices that…well don’t require a coherent, collimated, polarized light beam to operate. Or at least they weren’t until Peidong Yang’s came around with his nanolasers.

Unfortunately, all of these motors share the drawback of being unidirectional. It was until recently, with Ikeda’s et al. paper in Angew. Chem. Int. Ed. 2008, 47, 4986, that a very cool and new method for directly converting light into mechanical work. Basically they drew on the fact that azobenzene derivatives, when incorporated into liquid crystals, can have an isotropic phase transition induced isothermally by irradiation with UV light to cause trans–cis photoisomerization, and that the reverse transition can be induced by irratiation with visible light to cause cis-trans back-isomerization. This photoinduced phase transition
led to successfully reversible deformations of liquid crystal elastomers containing azobenzene chromophores just by changing the wavelength of the incident light.

Now this by itself doesn’t a motor make. There was one large problem: the liquid crystal elastomer had to be made into a film or “belt” for a motor. However, the LCE film by itself wasn’t mechanically strong enough and tended to crack after short light irradiation at high intensities. So to fix this issue, they simply laminated the LCE film with flexible polyethylene sheets. I love this type of simple solution to what could have been a convoluted problem. This is very much like what Mitch and I tend to do.

*Note that they did do a study of increasing light intensity and it’s correlation to the mechanical force generated by the film. They found that “the maximum force and the increment rate of the generated force are enhanced with an increase of the light intensity.”*

So what happened? Well check this out: Thats right. That is an actual light-driven motor NOT on the micro-scale. The diameter of those pulleys are 10mm on the left and 3 mm on the right. Sure it isn’t going to be competing in any races at the moment, but it could still be amazingly useful in the future. Light, straight to DC? That would be pretty darn awesome.

PS. Tomorrow is the first day of the experiment Mitch and I are running. Since we can, we will be broadcasting the first live cyclotron experiment out over the interweb. This may be one of the first live nuclear physics experiments broadcasted. Other then that, it is just cool. SO we will have it up all 24 hours as a “live vlog”.

Feel free; hell feel obligated to stop by, leave a comment, chat, ask questions, offer constructive or destructive criticism, whatever. Maybe ACS will pick this up as a new way to present new research: present it as it happens! Live!!!

By July 1, 2008 0 comments