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Hello everyone! I am back and today we continue the series of math questions. Today’s question is “A certain substance decomposes. Fifty percent of remains after 100 minutes.” Let’s see how to find the answer.

## Question: A certain substance X decomposes. Fifty percent of X remains after 100 minutes.

Here is question:

A certain substance X decomposes. Fifty percent of X remains after 100 minutes. How much X remains after 200 minutes if the reaction order with respect to X is (a) zero order, (b) first order, (c) second order?

In a first-order decomposition reaction, 50.0% of a compound decomposes in 10.5 min.

- a) What is the rate constant of the reaction?
- b) How long does it take for 75.0% of the compound to decompose?

## Answer

After being discharged into the drainage river, the organic pollutants in the urban wastewater will be decomposed by the bacteria present in the river water and this process is characterized by the biodegradation coefficient of organic substances K1s. dependent on water temperature, river flow velocity and other field factors.

with question a, the rate constant of the reaction is:

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With question b, time take for 75.0% of the compound to decompose is:

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## Relevent knowledge

The determination of the coefficients K1s is quite complicated, it depends on the concentration and characteristics of organic matter in the water, temperature, water flow velocity,… According to Rodzinler ID, 1984, [7], the coefficient K1s of river flow receiving wastewater is characterized by the expression:

K_{1s}= K_{1}xK_{d}

in there:

-K_{1}: biochemical oxidation rate constant of organic substances in the river water mixture taken at the first position with the condition of incubation at 20°C in the laboratory.

-K_{d}: coefficient that takes into account the actual conditions of the flow (watershed) studied: water temperature, flow velocity, etc.

The biodegradation of organic compounds by microorganisms follows a first-order kinetic equation, and the amount of biologically consumed dissolved oxygen yt after time t, mgL, is determined as follows:

y_{t}= L_{0}[1-exp(-k_{1}t)] or y_{t}= L0(1-10^{-K}1^{t})

in there:

-L_{0}: Initial BOD, mg/l ;

-k_{1} and K_{1:} BOD decomposition rate constant; k_{1} = 2,303 K_{1} , day^{-1.}

## Conclusion

The above article has helped you answer the question “A certain substance X decomposes. Fifty percent of X remains after 100 minutes”. Thanks for reading and see you next time!

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