The balloon crowd

First lets find the volume of the sphere with radius 10 cm.

vol. = 4/3 * pie * r^3.

substituting gives the volume as 4.1888 * 10^-3 cm^3

The density of an air molecule is given by: D = M/V.

Thus the volume of an air molecule

= M/D = 4.18 * 10^-23/1.23

= 3.9106 * 10^-23 cm^3

Therefore the number of air molecules in the sphere

= 4.1888*10^-3/3.9106 * 10^-23

= 1.0711 * 10^20 molecules

Hence the order of magnitude = 20

My method of approaching these types of problems is usually to state all the facts that we know/the facts that are given in the question. We know that this sphere has a radius of 10cm , filled with air. The air has a density of 1.23 kg/m^3 . After listing the facts, we now then start asking ourselves a series of questions.

To find out how many air molecules are inside the balloon, we first need to know what is the volume of air within this balloon. How can we find the volume of this balloon? Using the volume, is it possible to find the mass of the air? Using the mass, can we find the amount of molecules that constitutes that mass?

We can use the radius to find the volume. (To make things easier we'll convert the 10cm into r=0.1m , since we already know that the units used for the density is already in meters) Now using the formula for the volume of a sphere, with simply input the radius (in meters) in. $\begin{aligned} V_{sphere} =& \frac{4}{3}\pi r^3 \\ V_{sphere} =& \frac{4}{3}\pi (0.1 \mbox{ m})^3 \\ V_{sphere} =& 0.0041888... \mbox{ m}^3 \end{aligned}$

Now using the volume, we can find the mass of the air inside the balloon. $\begin{aligned} \mbox{mass} =& V_{sphere} \times \mbox {density} \\ \mbox{mass} =& 0.0041888 \mbox{ m}^3 \times 1.23 \frac{\mbox{kg}}{\mbox{ m}^3} \\ \mbox{mass} =& 0.0051512... \mbox{kg} \end{aligned}$

Now using the total mass and the mass of each molecule, we can find out many molecules there are. $\begin{aligned} \mbox{number of molecules} =& \frac{\mbox{mass}}{\mbox{mass of each molecule}} \\ \mbox{number of molecules} =& \frac{0.0051512 \mbox{kg}}{4.81 \times 10^{-23}\frac{\mbox{kg}}{\mbox{molecule}}} \\ \mbox{number of molecules} =& 1.07 \times 10^{20} \mbox{molecules} \end{aligned}$

Now with the number of molecules, what is the order of magnitude? Simply take the exponent of the number in scientific notation. $\boxed{20}$