Cocktail thermodynamics

I enjoy the occasional cocktail. My colleague Chris enjoys, makes and criticises cocktails. Door 34 in Bath makes really excellent cocktails. I have been known to indulge in a little thermodynamics; it was inevitable that one day these positions would collide in “the Boston Shaker Problem”. Chris swore on his reputation as a mixologist that:

  1. When cocktail ingredients are shaken with ice in a Boston shaker, the temperature drops to well below 0°C
  2. Using more ice results in less dilution
  3. Ice used in cocktail bars has been draining on the side, and is not below 0°C

This flies in the face of what we think we know about ice and water at atmospheric pressure, namely that:

  • If you attempt to raise the temperature of ice above 0°C, it will melt.
  • If you attempt to lower the temperature of water to below 0°C, it will freeze.
  • If you have a system of ice in contact with water, it will tend to equilibrate at 0°C, and resist changes to the temperature by melting and freezing as necessary.

If the ice starts at zero, and the other ingredients start above zero, where is the mechanism to lower the overall temperature? Seemingly the only available way is to take heat energy into the solid ice phase and melt. This isn’t going to work with non-melting ice cubes. It also doesn’t explain why we don’t hit our usual equilibrium temperature of 0°C.

The second part is just as confusing: there is a fixed amount of energy required to convert water between solid and liquid phases at a given temperature and pressure. This is known as the latent heat of fusion, \Delta H. This is an amount of energy per mass of water. As the total energy needed to cool your drink is fixed, the amount of ice that needs to be melted should also be fixed. (It’s not quite proportional, as you will need a little bit of extra melting to cool the extra water – and so on.) How does a greater bulk of ice help?
At this point we had a few ideas,  but turned to the internet for help – surely this is a well-known system? We did find an article at Cooking Matters, who not only hit on a good theory, but actually tested the effect with a thermocouple on their cocktail shaker. (It’s a pretty entertaining blog, if you’re interested in kitchen centrifuges and lobster-killing technique.) The key point is this: the system in a Boston shaker is not an ice-water equilibrium. The alcohol shifts the equilibrium to a lower temperature. Why? Entropy.

Door 34 asked for a mathematical description of the effect, and we gave them this:

\displaystyle 0 \geq \sum\limits_\textrm{mixer}^\textrm{ice} \left[\Delta U + \int mC_p \textrm{d}T \right] - \int S_\textrm{mixing} \textrm{d}T

It is worth giving a brief breakdown. The right-hand side of the equation is equal to a change in the Gibb’s free energy of the system. For a spontaneous process, this must be negative; at equilibrium it will equal zero (hence the ≥  sign.)  I’ve neglected the pressure-volume work as it is not expected to significantly contribute; strictly speaking there will be a little work associated with volume change as ice has higher specific volume than water, and water/ethanol mixtures have a lower specific volume than the sum of components.  I’ve also been a bit cheeky here in separating the vibrational contribution from the internal energy (strictly U should be U_{\textrm{pot}} or something.) This is to draw attention to Kirchoff-equation related goodness; the difference in heat capacities determines how much ice must melt. As the ice melts, \Delta U increases as water is a higher energy state than ice. Assuming no heat input, the enthalpy (in square brackets) must be conserved, and so the heat capacity C_p is integrated over a negative temperature difference to compensate; the temperature drops. This is driven by the entropy term on the far-right; as long as more entropy can be gained by melting ice to mix water and alcohol, the ice will continue to melt, dropping the temperature further, and well below 0°C. (Note that T in the equation is relative to absolute zero, and will never be negative.)

A problem remains– why does more ice mean less dilution? According to this, the equilibrium amount of melting is pretty much fixed. If anything, adding more ice should drive entropy in the direction of more melting. My best guess is that this is to do with the speed of the operation; a Boston shaker isn’t quite an isolated system. The longer it takes to reach equilibrium, the more heat will have entered the system and the “wetter” that equilibrium will be. This seems to agree with the account of mixologists, that shaking with more ice is a quicker experience that “feels” different. Still, I’m left feeling that this system isn’t quite solved yet…

Workflow: Chemical engineering undergraduate with a netbook

A few people have recently made remarks or asked about my netbook-based note-taking and workflow. I found a few blogs and articles helpful while developing my own system, so I thought I’d share the details here where other students or academics might stumble on them and find a useful idea or two.

Last year I finished an undergraduate master’s degree in chemical engineering. For the first three years of my degree I had trouble with note-taking and revision. The main causes of this were:

  • My own poor handwriting
  • The variety of formats in which we were provided with notes and resources
  • The ease of losing miscellaneous scraps of paper
  • Disagreement with lecturers on the best way to study their material

I was somewhat wary going into my fourth and final year that things were only going to get more difficult. It was time to sort this out. What I came up with was highly successful; not only did I have an excellent set of notes for revision, but I was able to work on design projects and coursework using a range of computers, fairly seamlessly.

Netbook: ASUS Eee PC 1001P

This was an affordable machine (I paid ~£240 for it in mid-2010) with a robust design, excellent battery life and sharp, matt 10″ screen. The screen is undermined by a distracting, glossy bezel, so I covered this with matt insulating tape. This has the bonus feature of covering the webcam when not in use. I tried to track down a netbook without Windows installed, but this didn’t appear to be an option for the current generation of netbooks.

Operating system: Ubuntu 10.04 Netbook Edition

Having dabbled with running linux in virtual machines on my iMac, I was keen to get stuck in with a more dedicated machine. Ubuntu Netbook Edition was an obvious choice, with a highly netbook-oriented user interface, but with the large community and package repositories of Ubuntu. I have resisted upgrading from version 10.04 as I understand there have been several UI changes, and I am very happy as it is. There were a couple of snags setting it up, most notably the erratic brightness control and unsupported wireless drivers. I found a very helpful blog, edited a configuration file, installed the wireless drivers with ndiswrapper, and I was up and running. I also encountered a few problems as a result of using wubi; while a nice toy for testing out Ubuntu on a windows machine, it is not very trustworthy for long-term use, and I would advise against it.

Once I had Ubuntu Netbook Edition up and running it has been a exceptionally smooth, consistent experience. The Software Centre is quite slick but I generally prefer to use apt-get from the command line. This post isn’t really the place to go into the relative merits of operating systems and package management, suffice to say that I currently plan to stick with Ubuntu and its spin-offs, largely due to the ease of installing software.

Annotating slides: Xournal

I try wherever possible to use software that is cross-platform. However, to the best of my knowledge, Xournal is only available on Linux and related platforms. It is a fairly simple package, designed for use with a stylus as a pseudo-notepad. However, it also offers very flexible PDF annotation. Where lecture notes have been made available in advance, I have converted them to PDF (if necessary) and opened them as a background in Xournal. From there I can type notes and sketch diagrams (usually in a contrasting colour and font) as though scribbling on a print-out, but with a few key advantages:

  • If the block of text is larger than expected, I can re-position or re-size it
  • Can correct mistakes cleanly
  • Can insert duplicate slides if need to take more notes than there is room for
  • Can insert blank slides to account for extended rants
  • No need to print in advance; can download and set up while in the lecture if necessary

The final step of the process is to export to PDF. From there, the notes can be studied from virtually any device.

Technical note-taking and report-writing: LyX

LyX is a free, cross-platform and user-friendly front-end for LaTeX. LaTeX is a typesetting system which is very popular among mathematicians and scientists due to its flexibility, power and relative ease in typesetting complex equations. However, working in pure LaTeX is rather intimidating. Here is a simple document prepared using LaTeX code, the LaTeX code used to generate it, and the LyX environment for the same piece of work:

LyX provides a more accessible interface for LaTeX, with many shortcuts for common functions, an accessible interface with plenty of clickable buttons, and you edit an abstract preview of the document. The spacing, font and page layout are not accurate, and should not distract you from the task at hand. Critically, maths and symbols are interpreted, so you can see what your equation will look like as you edit it. The shortcuts, while optional, are quick and ingenious. You have to see or try it in action, really – perhaps I will branch out into video and screen capture some day. By using LyX with keyboard shortcuts I was able to keep up with those lecturers who like to “chalk-and-talk” their derivations.

Diagrams: Inkscape and GIMP

Not much depth is needed here on this one; if you aren’t aware of these two programs you should probably check them out. They’re not quite as powerful as Adobe’s offerings, but they are capable of more than many people give them credit for. If you need to draw or trace a diagram, crop an image, remove some background elements and tweak the colour balance, these two programs will be more than sufficient.

Backup and file-sharing: Dropbox

The slight loss of control and security concerns are compensated by a very slick implementation. Essentially: there is a folder on your computer. The same folder is on any other computers you own. And the internet. It holds up to 2Gb (with various ways of expanding this). It’s a real folder, and you can use aliases, command line, file managers etc. however you like. Enjoy. In combination with other cross-platform software, this leads to many delightful situations where it simply does not matter which computer you are using.

General thoughts

My workflow has changed somewhat as a postgraduate student, but I maintain that this was a suitable and effective setup for an undergraduate engineer, and I would recommend it to others. Feel free to post below if you have any questions or comments! Finding the perfect workflow has become something of an obsession, and is analogous to the guitarist’s quest for “tone”. Hopefully some of the ideas here will help people on their own adventures.