In one kind of chemical reaction, unconverted reactants change into converted reactants. The fraction a of reactants that have been converted increases at a rate proportional to the product of the fraction of converted reactants and the fraction of unconverted reactants. Which equation describes this relationship? Choose 1 answer: (A) (da)/(dt)=(ka)/(1-a) (B) (da)/(dt)=ka^(2) (C) (da)/(dt)=ka(1-a) (D) (da)/(dt)=(k)/(a(1-a))

Question

In one kind of chemical reaction, unconverted reactants change into converted reactants. The fraction  a of reactants that have been converted increases at a rate proportional to the product of the fraction of converted reactants and the fraction of unconverted reactants. Which equation describes this relationship? Choose 1 answer: (A)  (da)/(dt)=(ka)/(1-a) (B)  (da)/(dt)=ka^(2) (C)  (da)/(dt)=ka(1-a) (D)  (da)/(dt)=(k)/(a(1-a))

Bytelearn · Accepted Answer

Denote Fraction of Reactants: Let's denote the fraction of reactants that have been converted by $ a $. According to the problem, the rate of change of $ a $ with respect to time $ t $, which is $ \frac{da}{dt} $, is proportional to the product of the fraction of converted reactants $ a $ and the fraction of unconverted reactants $ 1 - a $. The constant of proportionality is $ k $. Therefore, the equation that describes this relationship is $ \frac{da}{dt} = k \cdot a \cdot (1 - a) $.Rate of Change Equation: We can now compare the given options with our derived equation. Option (A) $ \frac{da}{dt} = \frac{ka}{1 - a} $ does not match because it suggests the rate is inversely proportional to the fraction of unconverted reactants, which is not what the problem states.Comparison with Options: Option (B) $ \frac{da}{dt} = ka^2 $ suggests the rate is proportional to the square of the fraction of converted reactants, which again is not what the problem states.Option (A) Analysis: Option (C) $ \frac{da}{dt} = ka(1 - a) $ matches our derived equation exactly, indicating that the rate of conversion is proportional to both the fraction of converted reactants and the fraction of unconverted reactants.Option (B) Analysis: Option (D) $ \frac{da}{dt} = \frac{k}{a(1 - a)} $ suggests the rate is inversely proportional to the product of the fraction of converted and unconverted reactants, which is incorrect according to the problem statement.

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