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Part
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How can I calculate total volume of a liquid when a powder is added to it? For example, if I add 1 cup of sugar to 1 cup of water, how can I calculate the resulting volume?
Hello and thank you for your question on how to calculate the resulting volume when adding powder to liquid. There is no short answer here so let's jump right into the calculations.
INTRODUCTION
Firstly it is important to understand that when you're looking at the chemistry of solutions there are endless factors to consider. Mixing between different liquids and solids vary depending on chemical structure, temperature, pressure, density and other properties (especially with water) that are unique to different compounds.
In this report I will show you how to determine an estimate that will yield a very close result, specifically for water and sugar, but will also hold true for most other mixtures.
METHODOLOGY
Let's have a look at the calculations.
The first and most important thing to get right are your units. One cup is equal to 8 fluid ounces. We can see from these tables that the packing density of granulated sugar is 881 kg/m3.
Using this conversion of 1kg/m3 : 0.0296 g/fl. oz. we can calculate the packing density of granulated sugar:
881*0.0296 = 26.054 g/fl. oz.
If one cup is equal to 8 fluid ounces, then the mass of the sugar can be determined:
26.054*8 = 208.434 g.
From these tables we see that the density of sugar is equal to 1586.2 kg/m3 (we are looking at the sucrose crystal value).
Converting these units to grams/fluid ounces:
1586.2*0.0296 = 46.909 g/fl. oz.
We can now calculate the volume of one cup of sugar without all the "wasted space" between the granules (packing as if the sugar is a liquid):
208.434/46.909 = 4.443 fl. oz.
Adding this value to the 8 fluid ounces occupied by the water alone gives us a total of 12.44 fluid ounces (8+4.44).
EXTENDING THE METHOD
If we had to generalize this method it would work as follows:
1. Find the packing density of the powder.
2. Packing density multiplied by volume gives you the mass of the powder.
3. The mass divided by the density of the powder gives you volume of perfect packing (as if it was a liquid).
4. Add this new found volume to the volume of the liquid to determine the total volume.
It is extremely important to always make sure that you are working with the right units. Furthermore, I highly recommend that you do a test round first. For instance if you have to heat up your liquid to dissolve all of your powder then these volumes might change. Sugar added to water hot enough to dissolve the sugar would not increase the total volume the same amount as if it is added to cold water.
CONCLUSION
Mixing one cup water with one cup sugar will increase the total volume of the cup of water but it won't double the volume, the volume will be an estimated 12.44 fl. oz. (not 16 as one might expect). An estimate of the new volume can be made by using the packing density, volume and density of the powder.