I keep a spreadsheet of the weekly concentration data from the Mauna Loa Carbon Dioxide Observatory, which is available going back to 1974. The first derivative of the concentration can be estimated by using a 52 week running average of changes over a ten year period, or at least this is how I estimate it. The first ten year period was available in 1984, and the first derivative so determined at that time was (14.06 ppm/10 years) = 1.41 ppm/year, in the week of May 24, 1984, when the concentration was 347.33 ppm. In roughly the same week of 2021, 37 years later, it was (24.43 ppm/10 years) = 2.44 ppm/year. Thus the second derivative is 1.03 ppm/year^2. If we integrate this and use the boundary condition of the value of the 1st derivative at just under 37 years to determine the integration constant, we have the first derivative. We repeat this integration to get the quadratic equation for concentration using the boundary condition of the value at May 23, 2021 which was 418.92 ppm.
Now, there is a lot of noise in these numbers, which is unsurprising, so the error bars are going to be very, very, very wide for using this crude equation.
To a very weak first approximation, a fatal concentration of carbon dioxide in air according to the CDC is around 100,000 ppm (0.1% in "percent talk" ), which is probably a function of Henry's law determinations of the adsorption coefficient of hemoglobin for CO2, and the more important action of carbonic anhydrase and other proteins which is the major mechanism for the physiological transport of CO2 in blood as bicarbonate. (Cf. H. M. Cheng and F. Jusof, Defining Physiology: Principles, Themes, Concepts, pg 107.)
However, this is an immediate toxic level, 100,000 ppm; the approach to this level may have profound physiological health effects. One might be very weak, even very sick, at half, or a quarter of this concentration over long periods.
But we could, in theory, if it wasn't silly, set the value at 100,000 ppm and solve the quadratic equation for the number of years.
But there are many other reasons why the quadratic equation is silly because it assumes fairly constant rates of accumulation over a very short period, 36-37 years. The first of these are the feedback loops, the melting of the permafrost and the vast number of forest fires we're seeing because too many of us believe that nuclear energy is "too dangerous," and climate change isn't "too dangerous." These factors are likely to overwhelm the numbers obtained from the Mauna Loa Observatory over this short period.
Then there is the issue of the volume of the oceans. The absorption of carbon dioxide by the oceans is a major constraint on atmospheric concentrations, and in fact, if one wanted to remove carbon dioxide from the air, one would be well advised to do it via the intermediate of seawater. (I think about this all the time.) To a first approximation, increased volume from melting ice should increase the overall capacity for CO2 uptake. It's not that simple though. The solubility of carbon dioxide is a function of its pH, but carbonic acid, formed by the hydration of CO2 is a weak acid. Thus the capacity of seawater to hold carbon dioxide to a first approximation is declining as the pH falls. But there is another factor. As the ice caps melt, the salinity of the ocean is changing, and therefore the concentration of critical salts for controlling carbonate concentrations, notably calcium and magnesium can be expected to decline, although acidification will also release CO2 found, for example, in crustacean shells. These are competing factors.
Magnesium oxide and carbonate are both fairly insoluble, but the oxide becomes more soluble in acidic conditions. Air pollution is adding both sulfuric and nitric acid to the oceans. (Magnesium sulfate is fairly soluble.) How this plays out is very complex, whether magnesium is leached from rocks by acid, or precipitated by carbonate.
There are dead zones in the ocean that occur naturally because of low iron concentrations, iron is important in many highly conserved metalloproteins, most importantly ferredoxin, a key enzyme in eukaryotic cell metabolism . Chlorophyll is a metalloprotein dependent on magnesium. Thus the efficiency of photosynthesis at sea, which provides most of the Earth's oxygen, is a function of magnesium concentrations, which in turn are functions of pH and carbon dioxide concentrations.
So the answer to your question seems to me at least, beyond my level of competence. This is a very complicated question.
The reality is however that we are doing nothing to address climate change other than to engage in bourgeois fantasies about wind turbines, electric cars, and solar cells. This naïve approach to this massive problem is beyond stupid.
I didn't watch the apparently satiric movie "Don't Look Up," but I heard about it. That's where we are. We are severely impacting the ecosystem in what in geologic time is a nanosecond; it's very much like a major meteor strike.
I think it didn't have to be this way, but it is. There will be hell to pay long before 100,000 ppm.
For the remainder of my overly long life however, I'll expect to hear again and again and again and again, right here at DU, that nuclear energy is "too dangerous." I will however be dead before these effects become inescapably obvious. Humanity is out of time, but I'm even more so.
I feel like I must be dreaming an awful dream, but I'm not.
Have a nice weekend.
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