If the rate with respect to \(\mathrm{O}_{2}, \mathrm{NO}\) and \(\mathrm{NO}_{2}\) are respectively $$ \frac{-\Delta\left[\mathrm{O}_{2}\right]}{\Delta \mathrm{t}}, \frac{-1}{2}, \frac{\Delta[\mathrm{NO}]}{\Delta \mathrm{t}}, \frac{+1}{2} \frac{\Delta\left[\mathrm{NO}_{2}\right]}{\Delta \mathrm{t}} $$ then the corresponding chemical equation is \(2 \mathrm{NO}\) \(+\mathrm{O}_{2} \rightarrow 2 \mathrm{NO}_{2}\)

We are given the rates of change for \(\mathrm{O}_{2}\), \(\mathrm{NO}\), and \(\mathrm{NO}_{2}\) as follows: 1. \(\frac{-\Delta\left[\mathrm{O}_{2}\right]}{\Delta \mathrm{t}}\) 2. \(\frac{-\Delta[\mathrm{NO}]}{2\Delta \mathrm{t}}\) 3. \(\frac{\Delta\left[\mathrm{NO}_{2}\right]}{\Delta \mathrm{t}}\)

We can use these rate expressions to determine the stoichiometric coefficients in the balanced equation. 1. For \(\mathrm{O}_2\) : The rate expression is \(\frac{-\Delta\left[\mathrm{O}_{2}\right]}{\Delta \mathrm{t}}\). The negative sign indicates that it is being consumed in the reaction, and its coefficient is 1 (since there is no other factor in the expression). 2. For \(\mathrm{NO}\) : The rate expression is \(\frac{-\Delta[\mathrm{NO}]}{2\Delta \mathrm{t}}\). The negative sign indicates that it is being consumed in the reaction, and its coefficient is 2 (since there is a factor of 2 in the denominator). 3. For \(\mathrm{NO}_2\) : The rate expression is \(\frac{\Delta\left[\mathrm{NO}_{2}\right]}{\Delta \mathrm{t}}\). The positive sign indicates that it is being produced in the reaction, and its coefficient is 1 (since there is no other factor in the expression).

Using the stoichiometric coefficients determined in Step 2, we can write the balanced chemical equation as: $$2 \mathrm{NO} \ + \ \mathrm{O}_2 \ \rightarrow \ 2 \mathrm{NO}_2$$