The correct rate constant in dm³ mol⁻¹ s⁻¹ is 4.8×1064.8×106, so option (B) is correct.​​

Introduction

Understanding how to convert a rate constant between different units is essential in chemical kinetics, especially when moving from molecular scale units to molar units used in laboratory calculations. In this article, the rate constant for the reaction O(g) + O₃(g) → 2O₂(g) is converted from cm³ molecule⁻¹ s⁻¹ to dm³ mol⁻¹ s⁻¹ step by step and every multiple-choice option is analysed.​

Step‑by‑step solution

Given:

Reaction: O(g)+O3(g)→2O2(g)Reaction: O(g)+O3(g)→2O2(g)

Rate constant:

k=8.0×10−15 cm3 molecule−1 s−1[][]k=8.0×10−15 cm3 molecule−1 s−1[][]

Goal:

k in dm3 mol−1 s−1k in dm3 mol−1 s−1

1. Convert cm³ to dm³

• 1 dm = 10 cm, so 1 dm3=(10 cm)3=1000 cm31 dm3=(10 cm)3=1000 cm3.​

• Therefore, 1 cm3=10−3 dm31 cm3=10−3 dm3.​

Apply this to the rate constant:

k=8.0×10−15 cm3 molecule−1 s−1×10−3 dm3 cm−3k=8.0×10−15 cm3 molecule−1 s−1×10−3 dm3 cm−3k=8.0×10−18 dm3 molecule−1 s−1[][]k=8.0×10−18 dm3 molecule−1 s−1[][]

2. Convert “per molecule” to “per mole”

• Avogadro constant NA≈6.022×1023 molecules mol−1NA≈6.022×1023 molecules mol−1.​

• 1 molecule−1=NA mol−11 molecule−1=NA mol−1.​

Multiply:

k=8.0×10−18 dm3 molecule−1 s−1×6.022×1023 molecules mol−1k=8.0×10−18 dm3 molecule−1 s−1×6.022×1023 molecules mol−1k=8.0×6.022×10−18+23 dm3 mol−1 s−1k=8.0×6.022×10−18+23 dm3 mol−1 s−1k≈48.176×105 dm3 mol−1 s−1=4.8×106 dm3 mol−1 s−1[][]k≈48.176×105 dm3 mol−1 s−1=4.8×106 dm3 mol−1 s−1[][]

So the converted rate constant is:

k=4.8×106 dm3 mol−1 s−1k=4.8×106 dm3 mol−1 s−1

Explanation of each option

Question options (as inferred from the image):

(A) 4.8×10−64.8×10−6 dm³ mol⁻¹ s⁻¹
(B) 4.8×1064.8×106 dm³ mol⁻¹ s⁻¹
(C) 4.8×10−94.8×10−9 dm³ mol⁻¹ s⁻¹
(D) 8.0×1068.0×106 dm³ mol⁻¹ s⁻¹

Option (A): 4.8×10−64.8×10−6 dm³ mol⁻¹ s⁻¹

• This value has the correct 4.8 coefficient but the exponent is negative, implying the rate constant becomes smaller after conversion, which contradicts the fact that switching from “per molecule” to “per mole” multiplies by Avogadro’s number (∼1023∼1023), making the numerical value larger.​

• Likely error: using 10+310+3 instead of 10−310−3 in the cm³ → dm³ conversion and then mishandling the exponent sign for Avogadro’s number.

Option (B): 4.8×1064.8×106 dm³ mol⁻¹ s⁻¹ (Correct)

• Matches the properly converted value obtained by multiplying by 10−310−3 for volume and 6.022×10236.022×1023 for molecules to moles.​

• Both the coefficient (≈4.8) and the exponent (+6) are consistent with dimensional analysis and standard conversion tables for rate constants.​

Option (C): 4.8×10−94.8×10−9 dm³ mol⁻¹ s⁻¹

• Again uses 4.8 but with an incorrect exponent; this corresponds to multiplying only by 106106 instead of by the full 10201020 that arises from 10−3×102310−3×1023.​

• Likely arises from using an approximate Avogadro’s number of 10231023 but then subtracting rather than adding exponents when combining 10−1810−18 and 10231023.

Option (D): 8.0×1068.0×106 dm³ mol⁻¹ s⁻¹

• Has the correct order of magnitude (106106) but wrong coefficient, suggesting someone multiplied by 10211021 instead of 6.022×10236.022×1023 or forgot the 10−310−3 factor from the cm³ → dm³ step.​

• This option is close numerically but does not follow from the exact conversion; therefore it is incorrect.

Key takeaways for exam preparation

• For rate constants expressed in cm³ molecule⁻¹ s⁻¹, multiply by 10−310−3 to change cm³ to dm³ and by NANA to change molecule⁻¹ to mol⁻¹.​

• Always handle powers of ten carefully: combining 10−1810−18 with 10231023 gives 105105, and the final coefficient must be calculated before adjusting to standard scientific notation.​