For More Information, we turn to our friends at Physics Central who figured out the specifics.
"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:
$latex Q=mc(T_f-T_i)&s=1$
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:
$latex Q=.25kg ast 4200(75_f-25_i)=.25kg ast 4200 ast 50C = 52,000 Joules&s=1$
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:
$latex Q=Pt&s=1&s=1$
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:
$latex t=Q/P&s=1$
$latex t=52000/.001=52000000seconds&s=1$
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"
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