Foundations of Quantum Mechanics and Physical Chemistry

Density is an intensive measure of “quantity”.  The quantity of a substance in Physical Chemistry is commonly measured by such extensive properties as weight, mass, volume, and intensive properties in concentration units, like Molarity (moles per liter),  Molality (moles per kg of solvent).  Whereas extensive properties are useful, concentration, or density, changes for different substances for the same measure: for example the number of dots per inch (dpi) is density just like the number of atoms per volume, etc.

Think of the analogy of buying food.  On the package you find not only the price, but also the price per kilogram.  The price per kilogram is more useful than the price alone, because it tells you the value of a kilogram of food and this allows you to compare prices.  It also allows you to mentally keep track of prices so you know when they change.  The price per kilogram is a property of what you are buying, whereas the price alone does not have that information.

Let us suppose that you read somewhere that perfect pancakes should have a density of 0.7 g ml^-1.  How could you ensure that you cook the pancakes so that they come out with that density?

If you weigh out the dry ingredients from your receipt, say flour, eggs, sugar, baking power, salt, etc. they will have a density of less than 0.7 g ml^-1.  They will be lighter, and therefore less dense than the final pancakes.  Water has a density of 1 g ml^-1, so what you need to do is to find the percentage of your dry ingredients so the density comes out right after cooking.  You can easily work out the density of the dry ingredients:

(1.1)

This is found with common kitchen measuring utensils, and can be expressed in units of g ml^-1.  First measure out 1,000 mls (one liter) of dry ingredients and weight them.  Let us suppose they weigh 500 g.  Hence the density is 500/1000 = 0.5 g ml^-1. If A is the percentage of water in the cooked pancakes, then the percentage of dry ingredients must be (100% -A), to give the final density.

(1.2)  

We need to determine A.  In Equation (1.2)  if A=100%, there is only water.  If A=0, then there are only dry ingredients.  So we can make batter in any proportion we want by solving for A after specifying the Final_Density(g ml^-1).

If  the Density(ingredients) is 0.5 g ml^-1, and we want the Final_Density to be 0.7 g ml^-1, then a quick calculation gives A = 4/10.  In this case, using Eq.(1.2), the cook can combine 4/10 x 1,000 g = 400 g of water with 6/10 x 1,000= 600 g of dry ingredients,

(1.3)

And we have 1,000 grams of pancakes (2.2 lbs) of the correct density.  If you want half that amount, the combination is mathematically equivalent,

(1.4)

Cooking, however, vaporizes water. Unless we know how much extra water to add, we cannot be sure the density will turn out right. We will have to be precision cooks because we have to know exactly how much water boils off during cooking.

For this, the batter must have more water than in Equation (1.2) so we change A to (A+V), where V is the percentage of liquid water that vaporizes due to cooking. We have already measured out the dry ingredients, and we know A=4/10, so all we need to know is the value of V. Then we prepare the batter to have a pre-cooked consistency of

(1.5)  

Clearly if the percentage of water, V, boils off, then after cooking, Equation (1.5) ends up at Equation (1.2)  with the desired density.

(1.6)  

The balance and measuring cups will have an error that you can determine by how accurately you can read the values.  For example the balance could be accurate to 10 g and the measuring cup to 1 ml for the ingredients and water, but you will need a more accurate graduated cylinder for the next step.  You can make a “cooking” standard by getting the frying pan to a fixed temperature for cooking.  Then measure out 10 ml ±0.5 ml of water at the same temperature as the batter. Add the 10 mls of water to the frying pan and time how long it takes to evaporate.  If it takes, say, 12 ±0.6 seconds to boil away 10 ml of water, then we make the assumption that the same volume of water evaporates in the same time from the batter.

It is possible to work out the new density of the batter as given by Eq (1.5), but it is not necessary. We simply prepare the batter with the desired density and amount wanted, Eq.(1.2) or Eq.(1.3), etc.  Then we determine how long to cook them.  For example if the cooking time is three minutes, the extra volume of water that must be added is (180 s x 10 ml/ 12 s) = 150 mls to the batter as prepared in Eq.(1.2). After cooking for three minutes, the pancakes will have the correct density of 0.7 g ml.

This makes me think this is the way that Sheldon from The Big Bang Theory might cook!

Cooking is, of course, chemistry. You might imagine companies that prepare cooked food in large quantities must do calculations like for our pancakes, and use precise amounts of ingredients and accurate cooking times so that their products are always consistent.

Do you use such precise calculations for your favorite recipes? Any thing to share?

EDIT (May 5th 2011): Thanks to Matt from sciencegeist.net for his additional helpful inputs (See comments below)

Pancakes Photograph by D. Sharon Pruitt