# Is mg/m^3 same as ppmw (CO2)?

I'm trying to convert $\ce{CO2}$ ppm (volume) in the air into ppm (weight) in the water.

I found this formula

$\mathrm{mg \ m^{-3}}$ = (ppmv)(MW) / [($0.08205$) (¦$\mathrm{K}$)]

where,

ppmv = air pollutant concentration, in parts per million by volume

$\mathrm{mg \ m^{-3}}$ = milligrams of pollutant per cubic meter of air

$\mathrm{K}$ = atmospheric temperature in degrees Kelvin = $273.15$ + $^\circ C$

$0.08205$ = universal gas law constant in $\mathrm{L\ atm \ mol^{-1} K^{-1}}$

MW = molecular weight of the air pollutant (dimensionless)

$\mathrm{atm}$ = absolute atmosperic pressure in atmospheres

$\mathrm{g \ mol}$ = gram mole

I've read the $\mathrm{mg \ m^{-3}}$ is a subunit of ppmw, so is it correct to say $\mathrm{mg \ m^{-3}}$ is ppmw?

For example, in a gas of very low density (e.g. $\ce{CO2}$ at 10-6 Pa), 1 mg/m3 could exceed the weight of $\ce{CO2}$, i.e. would be greater than 900,000 ppmw.