# Speed of sound - why does adding ethanol to water increase the speed of sound

1. Sep 21, 2012

### lavster

If you have a tank of water then the speed of sound is 1482 m/s. If you add ethanol, the speed of sound increases. This seems to be pretty linear with the % concentration of ethanol. why does ethanol (or propanol) increase the speed of sound?

thanks

2. Sep 21, 2012

### Bobbywhy

Can you please post the reference document for the above? This may help members in their effort to answer your query. Thank you.

Cheers,
Bobbywhy

3. Sep 21, 2012

### AlephZero

This seems rather strange.

Water: density 1000 Kg/m^3, bulk modulus 2.2 GPa
Speed of sound = sqrt(2.2/1.0) * 1000 = 1480 m/s, as you said.

Pure ethanol: density 789 Kg/m^3, bulk modulus 1.07 GPa
Soeed of sound = sqrt(1.07/0.789) * 1000 = 1164 m/s
which is slower than in water, not faster

4. Sep 22, 2012

### lavster

5. Sep 22, 2012

### AlephZero

I can see the graph, but I'm not going to pay money to read the paper - sorry!

6. Sep 22, 2012

### Bobbywhy

I also.

7. Sep 22, 2012

### Dickfore

I found the following link (see the section Bulk modulus of mixtures 1.6.2.1)
http://www.scribd.com/doc/51230908/46901640-Fluid-Mechanics#page=80

If you look at Eqn.(1.42), you will find that:
$$B_{\mathrm{mix}} = \frac{1}{\frac{x_1}{B_{1}} + \ldots + \frac{x_n}{B_n}}$$
where $x_i = V_i/V$ are the volume fractions of each component.

A similar derivation will convince you that the density of a mixture is:
$$\rho_{\mathrm{mix}} = x_1 \, \rho_1 + \ldots + x_n \, \rho_n$$

Now, suppose we have two liquids with volume fractions $x$, and $1 - x$. Then the speed of sound in the mixture is:
$$\begin{array}{l} c_{\mathrm{mix}} = \sqrt{\frac{B_\mathrm{mix}}{\rho_\mathrm{mix}}} = \sqrt{\frac{\frac{1}{\frac{x}{B_1} + \frac{1 - x}{B_2}}}{x \, \rho_1 + (1 - x) \, \rho_2}} \\ = \sqrt{\frac{B_1 \, B_2}{\left[ x \, B_2 + (1 -x) \, B_1 \right] \, \left[ x \, \rho_1 + (1 - x) \, \rho_2\right]}} \\ = \sqrt{\frac{B_2}{\rho_2} \, \frac{1}{ \left[ 1 + \left( \frac{B_2}{B_1} - 1 \right) \, x \right] \, \left[ 1 + \left( \frac{\rho_1}{\rho_2} - 1\right) \, x\right]}} \end{array}$$

At very low fractions x, we may expand this expression in powers of x, and the first two terms in the expansion are:
$$c_\mathrm{mix} = c_2 \, \left[ 1 + \left(-\frac{1}{2}\right) \, \left(\frac{B_2}{B_1} - 1 \right) \, x + o(x)\right] \, \left[ 1 + \left(-\frac{1}{2}\right) \, \left(\frac{\rho_1}{\rho_2} - 1 \right) \, x + o(x)\right]$$
$$c_\mathrm{mix} = c_2 \, \left\lbrace 1 + \left[1 - \frac{1}{2} \, \left(\frac{B_2}{B_1} + \frac{\rho_1}{\rho_2} \right) \right] \, x\right\rbrace$$
Now, in post #3, we were given the info:
$$\frac{B_2}{B_1} = \frac{2.2}{1.07} = 2.06$$
$$\frac{\rho_1}{\rho_2} = \frac{789}{1000} = 0.789$$
So, the coefficent in front of x is:
$$1 - \frac{2.06 + 0.789}{2} = 1 - 1.423 = -0.423$$

It corroborates the poster's conclusion that the mixture's speed of sound should decrease with volume fraction of ethanol.

If you have access to the paper, could you post the relevant plot where increase is shown?

8. Sep 22, 2012

### AlephZero

To see the plot, click the link in post #4. The values at the ends of the plotted range are
1519 m/s for 6% ethanol
1558 m/s for 12%

I might have used the wrong bulk modulus of ethanol.

http://www.efunda.com/materials/common_matl/show_liquid.cfm?MatlName=AlcoholEthanol gives 0.823GPa for ethanol at 20C and 2.18 for water at 20C.

But the numbers still don't agree. For 6% ethanol
Modulus: 1/B = 0.94 / 2.18 + 0.06 / 0.823
B = 1.983 GPa
density = 0.94 x 1000 + 0.06 x 789 = 987 kg/m^3
Speed of sound = sqrt (1.983/0.987) x 1000 = 1417 m/s
which is less than water, and not consistent 1519 on the graph.

9. Sep 22, 2012

### Dickfore

I have access to the paper. I don't know if I am allowed to upload the pdf.

Just to make sense of some of the number discrepancies we are getting, let me refer to their Table 1 for the temperature dependence of water in distilled water:

The cited source for the first column is:
V. A. Del Grosso and C. W. Mader, J. Acoust. Soc. Am. 52, 1442-1446 (1972)

The second column, of course, are the measurements by the authors of the paper we are considering.

Apparently the speed of sound in water is highly dependent on temperature.

From Fig.3 of the graph (the one linked in post #4), AlephZero deduced:
which would imply a speed of sound for pure water:

1519 + (0 - 6)*(1558-1519)/(12 - 6) = 1519 + (-6)*39/6 = 1480 m/s

Looking at the Table 1, this would correspond to a temperature:

19.9 + (1480-1481.4)*(19.9-18.0)/(1481.4-1475.1) = 19.9 + (-1.4)*1.9/5.7 = 19.9 - 0.47 = 19.4 oC

which is different from the quoted 20o C by 0.6 Co, a three times larger difference than the quoted temperature accuracy

Maybe they had some issues with temperature stability of the mixtures.

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10. Sep 22, 2012

### AlephZero

Sure, if only because the density variies significantly with temperature.

But it would be surprusing if the variation in speed of the mixsture went different ways at different temperatures. Fractions of a degree C shouldn't make much practical difference.

Do they give any properties for ethanol? With the numbers I used, the reduced speed matches "common sense". Addiong a small amount of ethanol reduces the bulk modulus a lot (reducing the speed a lot) and reduces the dessity a little (increasing the speed a little), so the net effect is a reduction.

I'm wondering what grade of ethanol those properties are for - absolute (abut 4% water content), anhydrous, or something else.

11. Sep 22, 2012

### Dickfore

This is a quote from their paper:
EDIT:
This may shed some light on the mystery:
Does anyone know what kind of a unit "v/v min" is?

Last edited: Sep 22, 2012
12. Sep 23, 2012

### AlephZero

I would take v/v as "volume / volume", i.e. the spec means "minimum 97.7% ethanol by volume", as opposed to "by mass".

The volume contraction effect they mention probably affects the bulk modulus of the mixture as well, which makes the short answer to the OP's question "The reason isn't simple".

The relationship can't be linear over the full range from 0% to 100% ethanol, unless somebody can make an argument that $c = \sqrt{K/\rho}$ doesn't apply to pure ethanol.

13. Sep 24, 2012

### lavster

wow - thanks everyone! glad im not missing something really obvious! your help is much appreciated!