# Technical Forum

## Effects of dilution on gravity and specific gravity

### Introduction

When a solution is diluted the only thing that changes is the volume. The mass of materials dissolved in the must or wine remains the same.

The mass, volume and concentration are related by the formula:

mass = volume x concentration

As the mass does not change the initial concentration times initial volume equals the final concentration times final volume or:

 ci x Vi = cf x Vf

ci is the intial concentration (g/L)
Vi is the initial volume (L)
cf is the final concentration (g/L)
Vf is the final volume (L)

It happens that the formula holds if gravity (G) is substituted for concentration. Recall that gravity equals the Specific Gravity - 1. If you double the volume you halve the gravity.

 Gi x Vi = Gf x Vf
Gi is the initial gravity or SGi -1
Gf is the final gravity or SGf -1

So if you have 3 L of must with an SG of 1.080 what happens if you make the volume up to 5 L?

 Gi x Vi = Gf x Vf since Gi = SGi -1 Gi = 1.080 -1 or 80 80 x 3 = Gf x 5 Gf = (80 x 3) / 5 Gf = 48 since Gf = SGf -1 SGf = 1.048
For the home fruit wine maker all is fine. For the commercial wine maker the story is not complete. Sugar and alcohol molecules hydrogen bond with the water molecules and the interspacial distances between the molecules change slightly with dilution or concentration. These slight volume changes affect the density (and hence gravity) and this is not accounted for by the above equations. For most of us the equation is just fine.

 Gi x Vi = Gf x Vf

How come the equation with G (dimensionless) instead of c (g/L) applies equally well. Curious to know?. Here is a proof in a round about way but its not for the feint hearted so feel free to ignore it and just use the above formula with confidence.

### A Proof

 SGi = mass volume Let X be the volume of water added. Now numerically the volume and mass of water is the same since the density of water is 1 SGf = m + X V + X m + X = SGf (V + X) SGiV + X = SGfV + SGfX (SGi - SGf)V = (SGf -1)X X = (SGi - SGf)Vi SGf - 1 Now we can arrive at the same equation starting with: Gi x Vi = Gf x Vf Vf - Vi = GiVi/Gf - Vi Gf(Vf - Vi) = GiVi - GfVi Gf(Vf-Vi) = Vi(Gi - Gf) (SGf - 1)(Vf - Vi) = Vi((SGi - 1) - (SG2 - 1)) V2 - Vi = X X = (SGi - SGf)Vi SGf - 1