# How do I convert kg·kg⁻¹ to ppbV (parts per billion volume)?

I am using the CAMS model output data to figure out the ground level ozone at a particular place. It is given in mass mixing ratio, and I want to convert to parts per billion volume mixing ratio (volume/volume). Shown above are ozone values from a model – basically the given values on the right beside the dates are in $\mathrm{kg} \cdot \mathrm{kg}^{-1}$. How can I convert them to ppbV?

• For clarification: you are looking for "ppbV" (parts per billion with respect to volume) and not for "ppbM" (... with respect to mass)? Do you have some other parameters given? Is it a homework question? What did you do to find the answer on your own? – daniel.neumann Feb 27 '18 at 16:44
• This is a VERY basic mathematical relationship unless I'm missing something, which I may be since you don't actually appear to ask a question in your question. – Ash Feb 27 '18 at 18:29
• As @daniel.neumann wrote, we need some more specifics here in order to give a useful answer: do you simply want the parts per billion of mass (in which case it's trivial, as per Ash's current answer), or a molar fraction, or a volume fraction, or something else? I'm going to vote to close this question temporarily as "unclear what you're asking"; please edit it to add the necessary information, and I will vote to re-open. – Pont Feb 27 '18 at 19:30
• @daniel.neumann Thank you for your response and my apologies for lacking clarity in my question! I am looking for ppbV. I shall explain what I'm trying to do. So I am using CAMS models' output data to figure out the ground level ozone at a particular place which is in mass mixing ratio and want to convert to volume mixing ratio/ volume itself. – HalloweenKing Feb 28 '18 at 2:19
• @Pont Good Day Pont! Apologies for the question being ambiguous. Yes, I am looking at the scenario based on the volume aspect/volume fraction, in my case ppb, since I would be looking at the pollutant concentration rather than simply jot down the mass of the pollutant component. Thanks! – HalloweenKing Feb 28 '18 at 2:30

So if you have a mass-mixing ratio, you effectively $\frac{ \text{kg pollutant}}{\text{kg dry air}}$. PPBV is parts per billion volume, or number of molecules of pollutant per billion molecules of dry air. Since not all particles weigh the same, you must use a conversion factor. Consider molar mass, which is the ratio of moles (a unit describing the number of molecules/photons/atoms) to mass. Ozone has a molar mass of 48 kg mol$^{-1}$. Dry air doesn't have a molar mass per se, but an effective molar mass may be computed by a weighted sum of the molar masses of the consitituent chemicals (or you can Google it). Afterwards, you may have to multiply by a billion.$$\therefore \frac{ \text{kg pollutant}}{\text{kg dry air}} \times\frac{\frac{\text{mols of pollutant}}{\text{kg of pollutant}}}{\frac{\text{mols of dry air}}{\text{kg of dry air}}}=\frac{\text{mols of pollutant}}{\text{mols of dry air}}$$