September 2nd, 2010

QotW #3: How Many Ice Cubes

Gaëlle enjoys soup but is impatient. Instant soups are great satisfiers of children and parents alike for their relatively good taste, variety and ease of preparation. The only problem still present is the impatient daughter part or how to cool the boiling soups rapidly, and relevant to this QotW, to the correct temperature. Ice cubes are the solution but how many should you put?

My Solution
My assumptions:

  • The soup is 250 mL (or grams) of mainly water at 99 °C
  • The 20 ml ice cubes are at 0 °C
  • The target soup temperature is 37 °C (body temperature)
  • We are at sea level
  • The soup is in an amazingly well insulated soup bowl with a lid
  • While we can assume that if we have one gram of water at 10 °C and one gram of water at 12 °C and mix them we will have two grams of water at 11 °C, the trick here is to figure out how much energy it takes to transform one gram of frozen water from a solid (ice) to a liquid. This is completely different than warming a liquid by one degree (assuming it is not reaching the boiling point). This has some important global implications regarding climate change. Once all the ice/glaciers have melted the additional heat added to the global system will have much greater impact than simply transforming solid to liquid – it will quickly heat the earths oceans dramatically increasing the number and intensity of hurricanes/typhoons etc…

    Back to the the problem at hand. We will solve this by calculating the energy required to bring the soup and an ice cube to 37 °C separately. Each ice cube weighs 18.4 grams (ice has 92% the weight of water – 0.92 * 20mL). We know that it takes 333.55 Joules of energy per gram of ice to transform it to liquid, and 4.18 Joules of energy per gram of water to raise the temperature by one degree. So if we want to melt 18.4 grams of ice:

    333.55 J * 18.4 g = 6137.32 J

    and heat it to 37 °C:

    4.18 J * 18.4 g * 37 = 2845.744 J

    It will require 8983.064 J to bring an ice cube to 37 °C. Note that it takes more than twice the amount of energy to melt the ice than it does to heat the water 37 degrees Celsius. Thinking back to the climate change topic we can see how quickly change will occur once the ice melts.

    To calculate the energy required to cool the cup of soup to 37 °C is simple in comparison:

    4.18 J * 250 g * (99 – 37) = 64790 J

    To calculate how many ice cubes we need we simply divide the cool down energy of the soup by the heating energy required by the ice cube:

    64790 ÷ 8983.064 = 7.21 ice cubes

    There you have it. If you have an amazingly insulated soup bowl and really impatient daughter, quickly place 7 ice cubes in the soup once it is ready. Obviously in the real world the soup cools much faster due to the surfaces it is in contact with. Ceramic soup bowls are great heat sinks. I usually only place one to two ice cubes for sufficient effect. Three if Gaëlle is extremely impatient.

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