Question Video: Identifying the Approximation Used in the Simplified Law of Dilution Chemistry

The equations below show a form of Ostwald’s law of dilution along with its simplified form: 𝐾_(𝑎) = 𝛼²/v(1 − 𝛼), 𝐾_(𝑎) = 𝛼²/v. Which of the following approximations results in the simplified form of the equation? [A] When 𝛼 is very small, (1 − 𝛼) can be approximated to be equal to 1. [B] When v is very large, (1 − 𝛼) can be ignored in the equation. [C] When 𝛼 is very large, (1 − 𝛼) can be approximated to be equal to 𝜙. [D] For strong acids, 𝛼 approaches 1, and so (1 − 𝛼) can be approximated to be equal to 𝜙.

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Video Transcript

The equations below show a form of Ostwald’s law of dilution along with its simplified form. Which of the following approximations results in the simplified form of the equation? (A) When 𝛼 is very small, one minus 𝛼 can be approximated to be equal to one. (B) When v is very large, one minus 𝛼 can be ignored in the equation. (C) When 𝛼 is very large, one minus 𝛼 can be approximated to be equal to zero. (D) For strong acids, 𝛼 approaches one, and so one minus 𝛼 can be approximated to be equal to zero.

Ostwald’s law of dilution can help us characterize how much a chemical species, such as a weak acid, dissociates into ions in solution. The law of dilution relates the acid dissociation constant of the weak acid to the degree of dissociation. The degree of dissociation is equal to the amount of dissociated substance divided by the total amount of substance.

For a weak acid, we can usually apply an approximation that can help us simplify the law of dilution. This is because the degree of dissociation is typically very small for a weak acid. If 𝛼 is very small, one minus 𝛼 will be approximately equal to one. If one minus 𝛼 is approximated to be equal to one, we will arrive at the simplified form of the law of dilution. This matches answer choice (A), but we can quickly confirm that the other approximations are incorrect.

Answer choice (B) states that when v is very large, one minus 𝛼 can be ignored. So, for example, let’s assume v equals 100 and one minus 𝛼 equals 0.8. Comparing the denominators of the two expressions, there is a large difference between 100 and 80. So, this wouldn’t be a very good approximation.

In answer choices (C) and (D), one minus 𝛼 is approximated to be equal to zero. If one minus 𝛼 was equal to zero, we will end up with zero in the denominator. So, the equation will be undefined.

So, the approximation that results in the simplified form of Ostwald’s law of dilution is statement (A). When 𝛼 is very small, one minus 𝛼 can be approximated to be equal to one.