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### A TeV, measured in chocolate and coffee

We toss around the term “TeV” – a teraelectron volt, 1012 electron volts (eV). But how much energy is it really?

An electron volt is the energy an electron gains when it is accelerated through a potential difference of one volt. An electron volt is defined as a unit of energy. (Various prefixes are defined here.)

Let’s put this in terms we can all understand. A Lindt 70% cocoa chocolate bar has 194 Calories. To convert this to electron volts:

194 Calories × 1000 calories/Calorie ×4.2J/calorie / 1.6 × 10-19 J/eV = 5 × 1024eV

(Note that the dietary unit, a Calorie, is 1000 calories, the amount of energy needed to raise the temperature of one mL of water by one degree Celsius.) So it would take a hundred billion (1011) proton-proton collisions at top energy (14 TeV in the center of mass) to get the same amount of energy as in a chocolate bar.

The difference is how much space we pack that energy into. A proton has a volume of roughly 1 fm3, or about 10-39 cm3. A Lindt chocolate bar is about 10 cm x 1/2 cm x 20 cm = 100 cm3. A chocolate bar then has an energy density of about 194 Cal/100 cm3, or around 2 Cal/cm3. A proton-proton collision at 14 TeV has an energy density around 14 x 1012 eV/10-39 cm3 x 1.6 × 10-19 J/eV *1000 Calorie/4.2J = 5 x 1035 Cal/cm3. So our proton-proton collisions have an energy density about 1035 times a chocolate bar.

We also use an electron volt as a unit of temperature. An atom in a monatomic (helium, argon, etc.) ideal gas has a kinetic energy of 3/2kBT where kB is the Boltzmann constant. The factor in front (3/2) is different for different systems. For instance, it’s 5/2 for a diatomic gas, such as hydrogen (H2), oxygen (O2), or nitrogen (N2). But the energy is usually kBT times some factor between 1-10. So to convert an electron volt into a unit of temperature, we use eV=kBT and T=eV/kB=11604 Kelvin.

So how hot is a cup of coffee in electron volts? When I worked at a coffee shop in high school, we made our cappuccinos and lattes at 160°F (71°C). This works out to be 344K, or 0.03 eV. So a proton moving at 7 TeV is about 100,000,000,000,000 (1014) times more energetic than the average molecule in a cup of coffee.

The Quark Gluon Plasma created at the Relativistic Heavy Ion Collider is at a temperature of about 170 MeV. (Note this is the temperature of the medium produced, not the energy of the incoming beam.) The fluid we’ll create at the LHC will be hotter – over ten billion (1010) times hotter than a cup of coffee.

These collisions are hot stuff!

[Note these are all what we call “back-of-the-envelope” calculations. The goal is to figure out the right order of magnitude for various quantities, not to do a detailed, precise calculation.]