# How long would you have to yell to heat a cup of coffee?

Yelling at a cup of coffee to warm it up? Does that even work? Believe it or not it absolutely does, it just takes a while.

Everything that happens, including warming up a cup of coffee or screaming at the top of your lungs, takes energy. What matters is how much energy there is and what form it all takes. When you yell, you convert the stored up (in physics speak its “potential”) chemical energy in your body into active (“kinetic”) sound energy.

The most important thing to remember that the amount of energy is always the same; it can never be spontaneously created or destroyed and thus has to come from somewhere. That way, if you trace it back, the chemical energy in your body came from the potential chemical energy stored in the nutritious bagel you ate for breakfast (essentially a calorie is the stored energy in food). That potential chemical energy in turn came from the wheat plant converting the radiant energy from sunlight into stored chemical energy. The sun shines because of nuclear reactions going on twenty four hours a day (and yes even at night) at its core.

So what does all this have to do with me hollering at my morning cup of Joe? It shows that once sound leaves your body it’s not the end of the story. Its energy still needs somewhere to go, so why not into the coffee? On a very tiny scale (as tiny as 1/100,000,000^{th} of a centimeter) everything is made of tiny vibrating particles called molecules. An object’s temperature is the measurement of how fast these molecules are moving around. The faster they’re moving the higher the temperature. Speeding up these molecules takes energy of course, and that’s where the shouting comes in.

Sound is just another form of energy, and one that isn’t too different from heat. Sound waves are the momentary compression and expansion of air molecules as they travel away from their source. What separates sound from heat is essentially how organized these vibrations are. Sound starts out as structured waves at its source that over time dissipates into the general disorder (“entropy”) of the other molecules in the atmosphere. The energy is still there, it’s just being bounced around molecules over a huge area that’s completely disorganized.

This dissipated sound energy bouncing around air molecules now takes the same form as heat. Both are exciting molecules, making them vibrate around faster and faster. So what happens when you add heat to something, it raises its temperature. “Temperature” is the measurement of how fast the molecules in a substance are moving, while “heat” is the net amount of thermal energy throughout the entire object.

Sound carries a relatively low amount of energy while warming up a cup of coffee an appreciable amount takes much more. The average human yells at about 80 decibels, which carries along with it about .001 watts of energy, about a 100,000 times less than the energy needed to light a standard 100 watt bulb. If you were to focus this energy at the average 8 oz cup of coffee (we’ll round up to .25 liters to make calculations easier) the average yell lasting a single second would warm the cup of coffee .00000095 degrees Celsius.

Calculating this is pretty simple to do once you have the raw data laid out. To find out how much energy it takes to change an object any number of degrees C you use:

Q = m · c · (T_{f} – T_{i})

Here, m is equal to the amount of mass of the object, c is the objects specific heat capacity which is a constant unique for every substance, the two Ts are the initial and final temperature and Q tells us how much energy in Joules we need all together.

There are a lot of implied assumptions when we do this “back of the envelope” calculation. (Personally I prefer using napkins.) We’re assuming things like a perfect transfer of energy from your yell to the coffee, a perfectly insulated cup that will never let any heat escape, and an unending even stream of energy. Obviously creating such a perfect system is extremely difficult in real life, but as we’ll soon see, there are lots of reasons why this method may not be terribly practical.

Now let’s start plugging numbers in. Starting out, say we wanted to raise the temperature of water from a tepid 25 C to a toasty 75 C. One quarter of a liter of water weighs .25 kg, and the temperature constant for water is 4200. Now we’re ready to revisit the equation:

Q = .25 kg · 4200 (75_{f} - 25_{i}) = .25kg · 4200 · ? 50°C = 52,000 Joules

Great! So what’s a joule? A joule is a unit of measurement that expresses how much power something consumes (or in this case would need to consume) during a period of time. It’s defined as:

Q = P · t

Where P is power and t is time. We know that we need 52,000 joules to heat the coffee, and that shouting at an average 80 decibels produces .001 watts of energy, we can rewrite the equation and plug everything in to find our answer:

t = Q / P

t = 52000 / .001 = 52,000,000 seconds

Which is 14,444 hours, 26 minutes and 40 seconds

Or 601 days and 20 hours 26 minutes and 40 seconds

In other words to heat up a quarter liter of coffee 50 C it would take:

1 year, 7 months, 26 days, 20 hours, 26 minutes and 40 seconds

That’s a long wait for your morning cup of Joe. I think I’ll stick it in the microwave for now.

Here’s a question though, how long might it take if you spoke softly to the coffee? I’ll get you started with a hint. The average person whispers at about 40 decibels, which translates out to about 10^{8} watts. You may notice that the power level of talking seems far lower than its decibel level might have you believe. This is because the power is on a logarithmic scale. One hundred decibels isn’t just one hundred times as powerful as a single decibel, but actually 10^{12} times as powerful. The difference in power between the lowest audible range and the level it can start to damage eardrums is so huge that decibels were invented so one didn’t have to mess with huge exponents when measuring how loud something is.

So knowing how many watts the average human voice is, how long would it take to heat the same cup of coffee 50° C if all you did was whisper sweet nothings at it?

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