If the concentration of #"SO"_2# in air is #"0.5 ppm"#, what is the percent by mass?

#0.00005%# #"w/w"# #"SO"_2# in #"air"#

This is basically a unit conversion. One way of defining a part-per-million, or #"ppm"#, is:

#"ppm" = "mg"/"kg"#

since there are #"1000 mg"# in a #"g"# and #"1000 g"# in a #"kg"#. In this case, you have:

#("0.5 mg SO"_2)/("kg air")#

The percent by weight, or weight percent, or #"wt%"#, or #"%w/w"#, is defined as:

#"%w/w" = (m_("component"))/(m_"total") xx 100%#,

where #m# is the mass in a given unit, such as #"g"# or #"mg"#, as long as both masses have the same units.

We just need to convert one of the units to the other. As the name suggests, #"1 kg"# is #1000000# times as large (physically) as #"1 mg"#. So, in a sense, a #"ppm"# is like a "percent", but #10000# times smaller (physically).

In other words, a #"ppm"# ("parts per million") is numerically #10000# times the weight percent, or a "parts per hundred", or there are #"10000 ppm"# for every #%#.

#"0.5 parts per million" xx ("1 part per hundred")/("10000 parts per million")#

#= color(blue)(0.00005% "w/w SO"_2 " in air")#