Density/ Scientific Notation

Question

What is the density, in kilograms per cubic centimeter, of gold and sodium together?

ELEMENT DENSITY( kilograms per cubic centimeter)

gold. 1.932 x 10^-2

sodium. 9.71 x 10^-4

Answer 1

David is absolutely right that the density depends on the relative amounts in the mixture, and the question as presented didn't state the relative amounts. However, assuming an equal mixture leads to the question, equal in what?

Maybe equal in mass, or maybe equal in volume. (His solution quietly assumed the latter. Also a glitch changing 4.855 to 4.655, which made only a small difference) And each of these requires the assumption of additive volumes in order to calculate the answer. i will present a solution for equal in mass, and one for equal in volume. To minimize use of exponents, I will work with large quantities, use g/cc instead of kg/cc, and then convert back at the end. So the densities will be 19.32 and 0.971 g/cc.

EQUAL IN MASS

1000 g Au and 1000 g Na. Calculate individual volumes. Since D = m/v, v = m/D.

1000/19.32 = 51.760 cc

1000/0.971 = 1029.866 cc

Total mass = 2000 g; total volume = 1081.626 cc

Combined density = 2000/1081.626 = 1.84907, round to 1.85 g/cc, or 1.85 X 10^-3 kg/cc

EQUAL IN VOLUME

1000 cc Au and 1000 cc Na. Calculate individual masses.

19,320 g Au

971 g Na

Total mass = 20,291 g; total volume = 2000 cc

Combined density = 20,291/2000 = 10.1455 g/cc = 1.015 X 10^-2 kg/cc.

Answer 2

The density of the mixture which contains both elements depends on the relative amount of each element in the mixture. If we assume an equal mixture of both elements, then we use a 1/2 factor to compute the weighted average for the density of the mixture.

weighted average for density = (1/2 * 1.932 x 10^-2) + (1/2 * 9.71 x 10^-4) = .966 x 10^-2 + 4.855 x 10^-4

The parentheses are optional. I am using them to highlight the 2 separate numbers which have different exponents..

I used 2 factors of 1/2 because there is an equal amount of both elements in the mixture

You can say there is 50% of both elements and 50% = .5 = 1/2.

.

If you add 1/3 of one and 2/3 of the other element, then the 2 factors would be 1/3 and 2/3.

You can use a calculator to do the addition. However, it is worth your time to go thru the process

one time to practice adjusting numbers in scientific notation.

Now, we need to adjust the 2 numbers and use the same power of 10 to add the numbers

The larger number determines the power of ten to use.

For example, if we add 100 and 1000, we use 1000 to determines the power of ten:

100 + 1000 = .1 x 10^3 + 1 x 10^3 = 1.1 x 10^3

First adjust the larger number to have a coefficient or mantissa between 1 and 10:

.966 x 10^-2 = (.966 x 10) * (10^-2 x 10^-1) = 9.66 x 10^-3

Notice how we multiple the coefficient by 10 to the power of 1 and the multiple the base by 10 to the power of -1. We need to always adjust both parts so the net effect is 10 to the power of zero.

Then we adjust the other number to use the same power of 10:

4.855 x 10^-4 = (4.655 * 10^-1) * (10^-4 * 10) = .4655 * 10^-3

Now that both numbers have the same power of 10, we can add the coefficients:

(.4655 * 10^-3) + (9.66 x 10^-3) = 10.1255 * 10^-3

Oops, our coefficient now is over 10 so we need to adjust the final answer:

10.1255 * 10^-1) * (10^-3 * 10^1) = 1.01255 x 10^-2

This is the final answer with units:

1.01255 x 10^-2 kg/cm^3

When you take a physics class, it will be very helpful and essential to think about the units

of numbers.