Physics Puzzler: Everything Is Better with S’mores
Whether it’s outside in the woods or in the comfort of their own living room, most people have had the experience of sitting near a warm fire, wanting something warm to eat. It is our opinion that those moments are best accompanied by indulging in a gooey, delicious snack—a s’more. And if you are anything like us, why waste a perfectly good opportunity to geek out? Just maybe we can make s’mores more awesome through science!
A s’more is simply a sandwich of marshmallow and chocolate, as seen in Fig. 1, with the marshmallow heated so that it partially melts the chocolate when they come in contact. For this puzzler we will not assume a spherical tasty treat, but we do need to define some of the components of this ideal physics s’more. Let’s assume a few constants:
Standard marshmallow
A standard marshmallow consists of mostly air by volume, with aerated sugar (C12H22O11), gelatin (long chains of amino acids), and water. Note: While at the time of printing there is not a NIST reference marshmallow, there is a NIST standard baking chocolate.1
Diameter: 0.025 m uncooked
Length: 0.038 m
Density: ~0.35 g/cm3
Melting point (sucrose): 186 °C
Specific heat2: 2.0 kJ/kg °C
Caramelization temperature: 160 °C
Graham cracker
Side: 0.064 m
Thickness: 0.0063 m
Chocolate
Assumption: Assume six chocolate pieces per s’more (of an average 12-piece chocolate bar—because...reasons)
Length of full bar: 0.136 m
Width of full bar: 0.054 m
Thickness: 0.006 m
Density: ~1300 kg/m3
Melting point ~30 °C (varies with type of chocolate)
Specific heat2, 3: ~1.8 kJ/kg °C
Thermal conductivity: 0.5 W m-1 K-1
Campfire
Average temperature: ~600 °C at the hottest point
Diameter: 0.5 m
Ambient temperature: 23 °C
Figure 1. An ideal, nonspherical s’more with a toasted marshmallow and a piece of chocolate sandwiched between two pieces of graham cracker.
Figure 2. A spherical cow roasting a nonspherical marshmallow over an ideal campfire (not to scale). Note that hot air and radiation are emitted from the campfire.
The first (and some say the most important) step for a s’more is roasting the marshmallow. The marshmallow can make or break a s’more, with the finer points having been debated at great length since a marshmallow was first roasted. The goal is to warm the marshmallow without it melting off the stick or burning it too much. To get it just right, you have to pay attention to where the marshmallow is held in relation to the fire, as seen in Fig. 2, and regulate the rate of heat transfer.
How much energy are we talking about? To calculate how much energy it would take to melt a marshmallow, we apply
Q = cmΔt (Eq. 1)
where Q is the heat needed, c is the specific heat, m is the mass of the marshmallow, and Δt is the change in temperature to melt the marshmallow. Applying our constants from page 12, we can see that it takes about 2.1 kJ, which is not a lot!
Whether it is preference or a complete accident, everyone has burned a marshmallow. Burnt marshmallow is simply a marshmallow that has been extremely toasted! There are two main processes that heat a marshmallow: absorption of campfire radiation (photons) and contact with very hot air rising off the fire (convection). If we place the marshmallow directly above the fire, we get both. This quickly heats the outside of the marshmallow, which causes the sugars in the marshmallow to break down and react. Some heat is absorbed inside the marshmallow, but it’s a slow process, as marshmallow is a good insulator.
Instead, let us consider trying to toast the marshmallow without completely burning it. The tricky part of this process is getting the perfect golden-brown exterior before the inside of the marshmallow melts completely and the whole thing falls off the stick into the fire in a blaze of glory. To achieve the perfect golden brown, it should be cooked slowly and indirectly, allowing the sugars to caramelize, making heavy use of the campfire’s infrared radiation while mostly avoiding the convection heat. As the outside of the marshmallow gets hot, the sugars/proteins begin to break down and then burn, which produces new flavors and smells, which gives s’mores their characteristic flavor4, 5 (and is also why everyone likes their marshmallow done a different way). It also forms a crust that helps the marshmallow keep its form. What makes the process a challenge is that caramelization of sucrose occurs near 160°C but the melting temperature of sucrose is 186°C, meaning you must keep your marshmallow within a rather narrow temperature range.
Once the ideal (melted and toasted) marshmallow is achieved, the next challenge is getting the chocolate to the perfect melted consistency in the sweet sandwich.
Compared to the marshmallow, chocolate has a much lower melting temperature, around 30°C. When you place the marshmallow in between the chocolate and graham crackers—without them it would be a mess—the marshmallow’s thermal energy is conducted into the chocolate. Assuming the marshmallow isn’t given time to cool after being heated, it should still have a temperature near its melting point of 186°C. We can use the heat transfer relationship for the total energy transferred Q through the chocolate:
Q = kAΔTt/L (Eq. 2)
where k is the thermal conductivity, A is the surface area, ΔT is the change in temperature, t is the time of the heat transfer, and L is the thickness.
Assuming that the chocolate starts at room temperature, to reach the melting temperature it will need to have a temperature change of ~7°C. We can also assume that we’re fairly impatient waiting for the s’more and only let it sit to melt for one minute. Using this information and the dimensions of the chocolate, the chocolate gains ~130 J, which is much less than the energy in the marshmallow! This means that the marshmallow can give off this much energy to the chocolate. But does the whole piece of chocolate melt?
The final part of the s’more is the graham cracker. Unlike the other components in the s’more, the graham cracker doesn’t go through any physical or chemical changes. Instead, it acts as an insulator for the sandwich, containing the heat to melt the treat within. It also doesn’t hurt that it keeps you from getting marshmallow all over your hands.
1) Prove to yourself that the chocolate needs about 130 J to melt using Eq. (2). How impatient could you be and still have it work? This assumes what about the temperature of the marshmallow?
2) Challenge: Now, ignore heat transfer rates and focus on just the total energy gains by the campfire. Assuming the graham crackers are perfect insulators and have a very small specific heat, what will the final temperature of the marshmallow and the chocolate ultimately be?
Check yourself online!6
References:
1. https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.260–176–2018.pdf .
2. American Society of Heating, Refrigerating and Air-Conditioning Engineers Handbook, Refrigeration, 2006.
3. P. Fryer and K. Pinschower, The materials science of chocolate, MRS Bulletin V25, 12, 2000; https://doi.org/10.1557/mrs2000.250 .
4. H. McGee, On Food and Cooking: The Science and Lore of the Kitchen, Scribner, 2004.
5. https://www.theverge.com/2017/6/11/15774634/marshmallows-smores-camping-camp-fire-summer-food-science .
6. https://www.spsnational.org/sites/default/files/sites/default/files/files/Puzzler_02202019.pdf .
