How much does butter shrink when frozen?

I just bought a butter dish which is a millimeter or two too small (the problems are that the corners are rounded which they forgot to mention before I ordered it and asked for the dimensions) to fit a piece of butter as it is sold in the supermarket. Since I don't want to return it my idea was to freeze the butter before I put it in this butter dish but for different reasons I can't make a "field test" just now (will update the question when I get the opportunity to do that). Instead I have to fall back to on theoretical answers.

How much does butter shrink when it's cooled from approximately 4°C (39°F) to -18°C (-0.4°F)?

Typically butter has about 1% salt, 82% fat and the rest is water.

• There is one clearly technically superior solution to freezing the butter: cutting the extra bit at the end, spreading it on fresh bread, and eating it!
– F'x
Jul 31, 2013 at 20:04
• @F'x, you read my mind, but I wanted to do the math to prove it. Jul 31, 2013 at 20:53
• Well, such a practical solution wouldn't be a good ground for a fun question at StackExchange, would it? :-) Besides, to be serious, I want to avoid greasing around with the butter before it is in the box. However, the butter is in fact 150 mm including the paper cover in the direction that is the bottleneck so 7‰ combined with removing the paper might be what does the trick. Otherwise, if it still does not work, Alessi should be ashamed of their industrial design of butter dishes.
– d-b
Aug 1, 2013 at 11:11
• The answer here would solve half of the problem. But the harder part is: how much does the butter dish shrink in the freezer? Oct 13, 2014 at 19:47

Given that fat (consisting mainly of assorted triglycerides of varying saturation) is the major constituent by a wide margin, it's probably reasonable to compare the change in density w/r/t temperature of butter to the change measured in various vegetable oils, since they have roughly comparable chemical composition (and because extensive data on them actually exists). I found a study in which the densities of various vegetable oils were measured over a range of temperatures. On average, the increase in density resulting from cooling from 110.0°C down to 23.9°C was roughly 6-7%. The data points graphing density against temperature arrange very linearly, and the regression lines generated fit them exceedingly well. I see no reason to suppose that the trend wouldn't continue as temperature decreases further. Assuming the comparison is reasonably valid, you could extrapolate that the increase in density you'd observe over the temperature range you specified should be on the order of approximately 2% at best.

As for the amount of shrinkage, strictly considering volume, the ratio of densities (assuming the 2% approximation is reasonable), would be:

$\frac{D_2}{D_1} = 1.02$

That is, the density after cooling, $D_2$, is 2% greater than $D_1$. That implies, holding mass constant:

$\frac{V_1}{1.02} = V_2$

Hence, given that volume is in units of cubic length, you can get a rough idea of how much the length of each side/edge should shrink by taking the cube root of both sides of the equation (obviously, this is only completely accurate for a perfect cube, but it emphasizes that the shrinkage in each dimension is smaller than the shrinkage in volume):

$\sqrt[3]{V_2} \approx 0.993\sqrt[3]{V_1}$

Unless your stick of butter is really gargantuan, a decrease of 0.7% on an edge is not going to account for 1-2mm.

Edit: Additionally, I've read that, in general, the magnitude of relative thermal expansion/contraction that liquids experience is usually greater than that experienced by solids, meaning that liquid density varies more over a given temperature range than solid density. Of course, I'll add the caveat: that's a very broad statement that admits of various exceptions. Nevertheless, this suggests that the estimate of 2% I made above by comparison to data on liquid vegetable oils is, if anything, more likely to be too large than too small.

• Would it be more efficient to heat up the stainless steel dish? Either with boiling water, in the oven or using my gas stove? How much does stainless steel expand when heated?
– d-b
Aug 1, 2013 at 15:30
• @user23122, I don't know off hand the degree to which it would expand, and I'm sure it varies with the exact composition. Unless you physicallly deform it, though, it will simply contract to its previous volume upon cooling. Aug 1, 2013 at 15:36
• @user23122, in general, I know that metals and alloys typically thermally expand much less than amorphous solids (e.g., plastics), and very much less than organic liquids. I think only inorganic salts and minerals are less apt to expand, but I could be mistaken. Aug 1, 2013 at 15:41