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Question

A scientist is studying the kinetic isotope effect in a reaction involving deuterium.  They find that the rate constant for the reaction with heavy water ($D_2O$) is significantly smaller than that with normal water ($H_2O$). This difference is primarily attributed to the:

P SRAVAN KUMAR · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
 **Primary Concept**: Kinetic Isotope Effect
 **Related Topics**: Reaction rates, vibrational frequencies, zero-point energy
 **Prerequisites**: Understanding of bond vibrations, quantum mechanics basics
## SOLUTION PATHWAY
**Given Information**:
  Rate constant for reaction with $D_2O$ is smaller than with $H_2O$.
  This indicates a kinetic isotope effect.
**Method & Steps**:
 1. Identify the primary cause of the kinetic isotope effect in this scenario.
  Reasoning: The kinetic isotope effect arises from the difference in the masses of isotopes, which affects the vibrational frequencies of bonds involving those isotopes.
  Formula/Rule:  The zero-point energy (ZPE) of a bond is given by $ZPE = \frac{1}{2}h
u$, where $h$ is Planck's constant and $
u$ is the vibrational frequency.  The vibrational frequency is inversely proportional to the square root of the reduced mass ($\mu$) of the atoms involved: $
u \propto \frac{1}{\sqrt{\mu}}$.
 2. Compare the vibrational frequencies and zero-point energies of O-H and O-D bonds.
  Working: Deuterium (D) is heavier than hydrogen (H).  Therefore, the reduced mass of the O-D bond is greater than that of the O-H bond.  Consequently, the vibrational frequency of the O-D bond is lower than that of the O-H bond.  This leads to a lower zero-point energy for the O-D bond.
  Key Point:  The reactants need to overcome the activation energy to proceed with the reaction.  A lower zero-point energy means the reactants involving deuterium have less energy initially, making it harder for them to reach the transition state and react.
 3. Conclude the primary reason for the observed difference in rate constants.
  Result: The difference in the zero-point vibrational energies of the O-H and O-D bonds is the primary reason for the slower reaction rate with heavy water.
  Verification: The other options are less significant factors in this context. Viscosity differences are relatively small, the dielectric constant difference is also not the primary driver of this kinetic isotope effect, and nuclear spin differences are not directly relevant to reaction rates in this case.
 ## OPTION ANALYSIS
A :x: Higher viscosity of $D_2O$ compared to $H_2O$ has a minor effect on reaction rates, but it is not the primary reason for the observed kinetic isotope effect.
B ✓: Difference in zero-point vibrational energies of the O-H and O-D bonds is the primary cause of the kinetic isotope effect. Heavier isotopes have lower vibrational frequencies and thus lower zero-point energies, leading to slower reaction rates.
C :x: While $D_2O$ does have a slightly different dielectric constant than $H_2O$, this difference is not the main factor affecting the reaction rate in this context.
D :x: The difference in nuclear spins of deuterium and hydrogen has a negligible effect on the reaction rate in this case.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Confusing kinetic isotope effect with other isotopic effects:  Remember that the kinetic isotope effect specifically deals with the rate of reaction.
  2.Ignoring the role of zero-point energy: The difference in zero-point energy is crucial in explaining the kinetic isotope effect.
 **Quick Tips**:
  1.Remember that heavier isotopes lead to lower vibrational frequencies and lower zero-point energies.
  2.Relate the zero-point energy difference to the activation energy barrier.

P SRAVAN KUMAR · Answer

Higher viscosity of $D_2O$ compared to $H_2O$.

P SRAVAN KUMAR · Answer

The correct Option is "Difference in zero-point vibrational energies of the O-H and O-D bonds."

P SRAVAN KUMAR · Answer

Greater dielectric constant of $D_2O$ compared to $H_2O$.

P SRAVAN KUMAR · Answer

Difference in the nuclear spins of deuterium and hydrogen.

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