Questions: (ln(high[OH^-])/ln(low[OH^-]))/(ln(slope of proper order plot with high[OH^-])/ln(slope of proper order plot with low[OH^-]))

The given formula is:

\[ \frac{\ln \left(\frac{\text { high }\left[\mathrm{OH}^{-}\right]}{\text {low }\left[\mathrm{OH}^{-}\right]}\right)}{\ln \left(\frac{\text { slope of proper order plot with high }\left[\mathrm{OH}^{-}\right]}{\text {slope of proper order plot with low }\left[\mathrm{OH}^{-}\right]}\right)} \]

This formula is used to determine the rate constant for a reaction based on the concentration of \(\mathrm{OH}^{-}\) and the slopes of the proper order plots.

• \(\text{high }[\mathrm{OH}^{-}]\): High concentration of \(\mathrm{OH}^{-}\).
• \(\text{low }[\mathrm{OH}^{-}]\): Low concentration of \(\mathrm{OH}^{-}\).
• \(\text{slope of proper order plot with high }[\mathrm{OH}^{-}]\): Slope of the rate law plot at high \(\mathrm{OH}^{-}\) concentration.
• \(\text{slope of proper order plot with low }[\mathrm{OH}^{-}]\): Slope of the rate law plot at low \(\mathrm{OH}^{-}\) concentration.

The formula can be interpreted as the ratio of the natural logarithms of the concentrations and the slopes:

\[ \frac{\ln \left(\frac{\text{high }[\mathrm{OH}^{-}]}{\text{low }[\mathrm{OH}^{-}]}\right)}{\ln \left(\frac{\text{slope of proper order plot with high }[\mathrm{OH}^{-}]}{\text{slope of proper order plot with low }[\mathrm{OH}^{-}]}\right)} \]

The formula for the rate constant for the complete rate law is:

\[ \boxed{\frac{\ln \left(\frac{\text{high }[\mathrm{OH}^{-}]}{\text{low }[\mathrm{OH}^{-}]}\right)}{\ln \left(\frac{\text{slope of proper order plot with high }[\mathrm{OH}^{-}]}{\text{slope of proper order plot with low }[\mathrm{OH}^{-}]}\right)}} \]