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

## Main Question or Discussion Point

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

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Bobbywhy
Gold Member
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
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

AlephZero
Homework Helper
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

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

Bobbywhy
Gold Member
I can see the graph, but I'm not going to pay money to read the paper - sorry!
I also.

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?

AlephZero
Homework Helper
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.

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:
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%
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

The temperature of the medium was controlled by a laboratory heater/stirrer and measured with a calibrated platinum resistance thermometer (ETI Ltd, Worthing, UK) with an accuracy of ±0.2°C.
Maybe they had some issues with temperature stability of the mixtures.

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

I'm wondering what grade of ethanol those properties are for - absolute (abut 4% water content), anhydrous, or something else.
This is a quote from their paper:
Measurements of c were made over a range of temperatures in mixtures of ethanol (99.7% v/v min, Hayman Ltd, Witham, Essex, UK) in freshly distilled water. Ethanol concentrations of 6%, 8%, 10% and 12% by volume were used. The estimated uncertainty in the concentration was ±0.1%.
EDIT:
This may shed some light on the mystery:
Volume %

The Volume % is bit strange in that the Volume % of ethanol and the Volume % of water add up to more than 100. The way the Volume % of ethanol is defined is that it is the parts of ethanol to which water has been added to bring the volume to a total of 100 parts.

For example, when working at 20°C, if you took 48.00 liters of pure ethanol you would find that you would have to add 55.61 liters of water to bring the total volume to 100.00 liters. The Volume % of ethanol in this case is 48.00% and the Volume % of water is 55.61%. The total of the separate components is 103.61 liters, but has contracted to 100.00 liters.

The degree of contraction is affected by the relative quantities ot ethanol and water, and also by the temperature. For example, when working at 50°F you would find that if you again started with 48.00 liters of ethanol you would need 55.82 liters of water to bring the total to 100.00 liters. This means that when a strength is specified in Volume % the temperature has to be specified as well.

Ethanol has a significantly higher coefficient of expansion than water. This means that as the temperature varies, the volume of the ethanol portion of the mixture changes faster than that of the water portion.
Does anyone know what kind of a unit "v/v min" is?

Last edited:
AlephZero
Homework Helper
Does anyone know what kind of a unit "v/v min" is?
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.

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