Question

In nmr experiments, structural inferences sometimes are drawn from differences in resonance frequencies as small as \(1 \mathrm{~Hz}\). What difference in energy in kcal \(\mathrm{mol}^{-1}\) does \(1 \mathrm{~Hz}\) represent?

Short answer

The energy difference for a 1 Hz frequency is approximately \(1.584 \times 10^{-37}\) kcal/mol.

Step-by-step solution

Understand the Relationship between Frequency and Energy

The energy difference between two states in nuclear magnetic resonance (NMR) can be calculated using the formula \( \Delta E = h \times u \), where \(h\) is Planck's constant \( (6.626 \times 10^{-34} \text{ J s}) \) and \(u\) is the frequency difference in Hz.

Calculate Energy Difference in Joules

First, we calculate the energy difference (\(\Delta E\)) for a frequency difference of \(1 \text{ Hz}\): \[ \Delta E = 6.626 \times 10^{-34} \text{ J s} \times 1 \text{ Hz} = 6.626 \times 10^{-34} \text{ J} \]

Convert Joules to kcal/mol

To convert the energy from Joules to kcal/mol, we use the conversion factor \(1 \, \text{J} = 2.390 \times 10^{-4} \, \text{kcal/mol}\): \[\Delta E = 6.626 \times 10^{-34} \text{ J} \times 2.390 \times 10^{-4} \frac{\text{kcal/mol}}{\text{J}} \approx 1.584 \times 10^{-37} \text{ kcal/mol}\]

Key concepts

Energy Difference Calculation

In the realm of nuclear magnetic resonance (NMR) spectroscopy, understanding the energy difference between various energy states is crucial. This difference is what allows scientists to infer structural information from the spectra. The fundamental relationship used here is expressed with the equation \( \Delta E = h \times u \), where \( \Delta E \) is the energy difference, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J s}) \), and \( u \) is the frequency difference in Hertz (Hz).

To calculate the energy difference, you simply multiply Planck's constant by the frequency difference. In practical terms, even a small frequency difference, like \(1 \text{ Hz}\), can represent a tangible energy difference. This scalability makes NMR a powerful tool for examining intricate molecular structures.

This calculated energy change can illustrate how sensitive NMR spectroscopy truly is and why it's effective for identifying minute structural differences in compounds.

Frequency and Energy Relationship

Frequency and energy are interconnected in the world of physical chemistry and NMR spectroscopy through Planck's relation. The equation \( \Delta E = h \times u \) elegantly demonstrates how energy and frequency are related. In this equation, \( h \) (Planck's constant) is the proportionality factor that bridges energy and frequency.

Increasing the frequency results in a proportionate increase in energy. This relationship is linear, meaning that as frequency doubles, so does the energy difference. Understanding this direct proportional relationship helps in predicting how changes in resonance frequencies can affect the energy state of a molecule.

Such predictions are instrumental for chemists who rely on detailed frequency shifts to discern subtle structural and electronic changes in molecules, extracting information that might otherwise remain concealed.

Unit Conversion in Chemistry

Often in chemistry, converting units is necessary to translate scientific findings into practical understanding. In NMR spectroscopy, once the energy difference is calculated in Joules from a frequency difference, it often makes more sense to express this energy in terms of kcal/mol, a common unit in chemistry.

The conversion utilizes a known factor: 1 Joule equals approximately \(2.390 \times 10^{-4} \text{ kcal/mol}\). Using this conversion factor, scientists can understand and communicate energy differences in a more familiar scale. For example, converting \(6.626 \times 10^{-34} \text{ J}\) to kcal/mol allows researchers to appreciate the magnitude of energy changes in a relatable form.

Unit conversion is crucial not only to maintain consistency between different chemical measurements but also to aid in the collaborative nature of scientific research where results are shared across various disciplines and regions. Mastery of this skill ensures that any calculated energy difference is both usable and understood easily across different contexts.