JEE Main 2021: Viral Inactivation Rate Constant

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Daily Challenge JEE Main 2021 (20 Jul, Shift 1)

Viral Inactivation Kinetics

Deduce the order of reaction from the problem statement and apply logarithmic properties.

Question:

The inactivation rate of a viral preparation is proportional to the amount of virus. In the first minute after preparation, $10\%$ of the virus is inactivated.

The rate constant for viral inactivation is ________ $\times 10^{-3} \text{ min}^{-1}$.
(Nearest integer)

[Use: $\ln 10 = 2.303 \,;\, \log_{10} 3 = 0.477 \,;\, \text{property of logarithm : } \log x^y = y \log x$]

Enter your integer answer:

Detailed Solution

Concept Alert: Deciphering the Order
The problem states: "The inactivation rate... is proportional to the amount of virus."
Mathematically, $\text{Rate} \propto [Virus]^1$. This is the exact definition of a First-Order Reaction! You must deduce this to know which formula to use.

Step 1: Set up the concentrations

For a first-order reaction, the integrated rate law is:

$$ k = \frac{2.303}{t} \log \left( \frac{[A]_0}{[A]_t} \right) $$

Given values:

• Let the initial amount $[A]_0 = 100$
• Time $t = 1 \text{ min}$
• $10\%$ is inactivated, so amount reacted $= 10$.
• Amount remaining, $[A]_t = 100 - 10 = \mathbf{90}$

Step 2: Substitute and Expand the Logarithm

Substitute the values into the equation:

$$ k = \frac{2.303}{1} \log \left( \frac{100}{90} \right) = 2.303 \log \left( \frac{10}{9} \right) $$

Use the property $\log(\frac{a}{b}) = \log a - \log b$:

$$ k = 2.303 \left( \log 10 - \log 9 \right) $$

Step 3: Utilize the Given Log Properties

We know $\log 10 = 1$. For $\log 9$, we can rewrite $9$ as $3^2$ and use the property given in the hint: $\log x^y = y \log x$.

$$ \log 9 = \log(3^2) = 2 \log 3 $$

Substitute the given value of $\log 3 = 0.477$:

$$ k = 2.303 \left( 1 - 2(0.477) \right) $$ $$ k = 2.303 \left( 1 - 0.954 \right) $$ $$ k = 2.303 \times 0.046 $$ $$ k \approx 0.105938 \text{ min}^{-1} $$

The question asks for the answer in the format $Y \times 10^{-3} \text{ min}^{-1}$.

$0.105938 = \mathbf{105.938} \times 10^{-3} \text{ min}^{-1}$

Rounding to the nearest integer gives $106$.

Final Answer: 106

Review First Order Kinetics

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