Computing the concentration of a reactant given formation of products

At high temperatures nitrogen dioxide decomposes into nitrogen oxide and oxygen gas according to $$\ce{2 NO\_2 -> 2 NO + O\_2}$$


I have a question now. Let's say at a certain temperature I'm given data for the formation of $\ce{O\_2}$. For example:
At $t = 0$ we have $[\ce{O\_2}] = 0$ mol/L, at $t = 50$, $[\ce{O\_2}] = 7.10 \cdot 10^{-4}$ mol/L, at $t = 100, [\ce{O\_2}] = 1.205 \cdot 10^{-3}$ and so on until a certain time.


If I'm given that the initial concentration of $\ce{NO\_2}$ is $0.006M$, how can I know what the concentration of $\ce{NO\_2}$ is at certain times? I'm asked to determine the order of the reaction on the basis of the given data, and also to present graphically the rate constant.


To do that I need to know if $\ln([\ce{NO\_2}])$ versus time $t$ is a straight line. But how to find $[\ce{NO\_2}]$ in the first place?




In this reaction, the production of $\ce{O2}$ is tied to the decomposition of $\ce{NO2}$; for each molecule of $\ce{O2}$ that is produced, there have to be 2 molecules of $\ce{NO2}$ that have decomposed. So:


$$2 \Delta n\_{\ce{O2}} = -\Delta n\_{\ce{NO2}}$$


where $\Delta n$ represents a *change* in the number of moles. The question does not specify, but I will assume that the volume of the reaction is kept constant; since concentration is amount divided by volume, this means that the same relation holds for concentrations:


$$2\Delta [\ce{O2}] = -\Delta [\ce{NO2}]$$


Since the initial concentration of $\ce{O2}$ is zero, $\Delta [\ce{O2}] = [\ce{O2}]$, and therefore


$$2[\ce{O2}]\_t = [\ce{NO2}]\_i - [\ce{NO2}]\_t $$


$$[\ce{NO2}]\_t = [\ce{NO2}]\_i - 2[\ce{O2}]\_t$$


To use the numbers in the question, at $t = 50$ s we have $[\ce{NO\_2}] = (0.006 - 2(7.10\cdot 10^{-4}))$ M = $4.58\cdot 10^{-3}$ M.


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