What if your cocoa went in a flask?

A grown-up pours you a mug of hot cocoa. Yum!

But wait a little while, and it gets cool. What if the cocoa went in a flask instead?

A mug of hot cocoa never stays hot for long. Leave it on the table and in a few minutes it is warm, and later it is cold. The heat in the cocoa is slipping away into the room.

A vacuum flask looks like an ordinary bottle, but people carry hot drinks in one for a whole day out. What if your cocoa went in a flask? In this lab you can pour it into a mug, a mug with a lid, or a flask, and watch the thermometer.

Heat flows from a hot object to cooler surroundings until they reach the same temperature. A drink cools fastest when the difference is large and slows as it approaches room temperature.

This lab compares containers using one number, the cooling constant k: the fraction of the drink-to-room temperature gap lost each minute. It comes from Newton's law of cooling, which says an object cools at a rate proportional to the difference between its temperature and that of its surroundings (OpenStax Calculus, §6.8). The question is how much a vacuum flask changes k, and what that means for how long cocoa stays hot.

In Olly's lab, the cocoa starts very hot. The room is just warm.

In an open mug, the cocoa cools down fast. After an hour, it is nearly as cool as the room.

In the model, the cocoa starts at 80 °C in a room at 20 °C. In an open mug it cools quickly at first: after 10 minutes it is down to about 62 °C, and it reaches 60 °C after about 12 minutes.

Then the cooling slows down, because the cocoa is not so much hotter than the room anymore. After a whole hour the open mug is at about 27 °C, only a little warmer than the room. Put a lid on and it does better: still about 40 °C after an hour.

With T₀ = 80 °C, T_room = 20 °C and k = 0.035 per minute, the model gives T(t) = 20 + 60·e^(−0.035t). After 10 minutes T ≈ 62.3 °C; it crosses 60 °C at 11.6 minutes; and the 60-degree gap halves every ln 2 ÷ k ≈ 19.8 minutes. After 60 minutes the mug is at 27.3 °C.

A lid lowers k to 0.018 per minute in the model: the gap now halves every 38.5 minutes, the cocoa takes 22.5 minutes to reach 60 °C, and it is still at 40.4 °C after an hour.

Now Olly pours the cocoa into a flask.

The cocoa stays hot much longer! After an hour, it is still nice and hot.

Now pour the same 80 °C cocoa into a vacuum flask. In the model it cools so slowly that after 10 minutes it is still about 78 °C. It takes about 101 minutes, well over an hour and a half, to reach 60 °C, and after a whole hour it is still about 67 °C.

The open mug lost more than 50 degrees in that hour. The flask lost only about 13. Try a freezing day too: set the room to 0 °C and the flask's cocoa is still about 63 °C after an hour.

With k = 0.004 per minute the model's gap halves only every 173.3 minutes, almost three hours, nearly nine times longer than the open mug. After 10 minutes the cocoa is at 77.6 °C, it reaches 60 °C after 101.4 minutes, and at the end of the 60-minute run it is at 67.2 °C.

The shape of the curve is the same exponential decay; only the rate changes. Lower the room to 0 °C and the flask still reads 62.9 °C after an hour, while an open mug falls to 9.8 °C.

Heat sneaks out of the mug in three ways. It goes through the cup, into the moving air, and out as invisible rays.

A flask has two walls with almost no air between them. Heat can hardly get across.

Heat leaves a hot drink in three ways. Conduction passes heat through things that touch, like the mug's wall. Convection carries heat away in moving air. Radiation sends heat out as invisible rays.

A vacuum flask has two walls with the air pumped out of the gap between them. With almost nothing in the gap, heat can hardly be conducted or carried across. The walls are silvery, which sends back the heat rays. Our model sums all of this up in one cooling number, and it leaves out evaporation (steam), which makes a very hot drink cool faster at first.

Conduction is heat transfer through stationary matter by contact; convection moves heat with a flowing fluid; radiation carries it as electromagnetic waves. A vacuum flask attacks all three: the evacuated gap between its walls is, in Britannica's words, practically a nonconductor of heat, and with no air there is nothing to carry convection currents. Silvering cuts radiation, because a silver surface is a poor absorber and a poor emitter.

Model limits: Newton's law of cooling assumes the heat-loss rate is proportional to the temperature difference, with one constant k. The k values for mug, lid and flask are rough typical values, not measurements of real cups, and evaporation, which makes a very hot drink cool faster at first, is left out.

A scientist named James Dewar made the first vacuum flasks.

He wanted to keep very, very cold liquids cold. A flask keeps cold things cold, and hot things hot!

The vacuum flask was invented in the 1890s by a British scientist, Sir James Dewar. He wasn't thinking about cocoa: he wanted to store liquefied gases, which are extremely cold, by stopping heat from the room getting in.

The same trick works both ways: it keeps cold things cold, and it keeps a hot drink hot for several hours. The weak spot is the neck, the only place where the inner and outer walls join, so flask makers keep it as narrow as they can. Thermos is a brand name for a flask in a metal case.

Britannica describes the vacuum flask as a vessel with double walls, glass or steel, with the space between them evacuated. Sir James Dewar devised it in the 1890s to preserve liquefied gases by preventing heat from the surroundings reaching the liquid. Radiation is minimized by silvering the walls.

The main remaining heat path is the neck, the only junction between the walls, which is therefore made as small as possible; Britannica notes that a hot liquid stays at a high temperature in a flask for several hours. Our model's single k = 0.004 per minute captures that slow loss overall but not where it happens.