Question

Calculate the amount of energy required to spin-flip a proton in a spectrometer operating at \(300 \mathrm{MHz}\). Does increasing the spectrometer frequency from 200 to \(300 \mathrm{MHz}\) increase or decrease the amount of energy necessary for resonance?

Short answer

The energy required for a 300 MHz spin-flip is \(1.9878 \times 10^{-25} \text{J}\). Increasing the frequency increases the energy needed.

Step-by-step solution

Understand the Formula for Energy

The energy required for a spin-flip, especially in an NMR spectrometer, is given by the formula: \( \Delta E = h u \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{Js} \) and \( u \) is the frequency of the spectrometer. In this case, \( u = 300 \text{MHz} \).

Calculate the Energy for Spin-Flip

Substitute the frequency \( u = 300 \times 10^6 \text{Hz} \) into the formula. Calculate \( \Delta E = 6.626 \times 10^{-34} \text{Js} \times 300 \times 10^6 \text{Hz} \).

Perform the Calculation

Multiply the values: \( \Delta E = 6.626 \times 10^{-34} \times 300 \times 10^6 = 1.9878 \times 10^{-25} \text{J} \). This is the energy required for the spin-flip in a 300 MHz spectrometer.

Compare Energy Requirement at Different Frequencies

Repeat the calculation for a 200 MHz spectrometer by using \( u = 200 \times 10^6 \text{Hz} \). Calculate: \( \Delta E = 6.626 \times 10^{-34} \times 200 \times 10^6 = 1.3252 \times 10^{-25} \text{J} \).

Analyze the Effect of Increased Frequency

By comparing the two results, the energy for the 300 MHz spectrometer (\(1.9878 \times 10^{-25} \text{J}\)) is greater than that for the 200 MHz spectrometer (\(1.3252 \times 10^{-25} \text{J}\)). Increasing the frequency from 200 MHz to 300 MHz increases the amount of energy necessary for resonance.

Key concepts

Spin-Flip Energy

The concept of spin-flip energy is crucial when understanding NMR spectroscopy. In the context of NMR, protons in a magnetic field can exist in different energy states. To jump from a lower to a higher energy state, they need an external energy input, called the spin-flip energy.
This energy is specifically applied to flip the spin of the proton. Think of it as nudging a spinning top gently enough to make it flip on its side.
The amount of spin-flip energy depends on the strength of the magnetic field and the frequency of the spectrometer used.

• Higher frequencies in spectrometers imply a stronger magnetic field.
• More energy is required to achieve resonance at these higher frequencies.

In essence, knowing the spin-flip energy gives insight into the magnetic environment of the nucleus being studied.

Planck's Constant

Planck's constant is a fundamental part of quantum mechanics, symbolized as \( h \). It connects the energy of a photon to the frequency of electromagnetic radiation via the equation \( E = hu \).
In NMR spectroscopy, this constant ensures that we can calculate the exact amount of energy needed to flip the spin of nuclei, such as protons.

• Planck's constant has a value of \( 6.626 \times 10^{-34} \text{Js} \).
• It bridges the physical phenomena of energy and frequency in the quantum realm.

If you picture energy as currency, Planck's constant is a universal exchange rate indicating how much energy corresponds to one cycle of light at a given frequency. Understanding this concept is essential for calculations in NMR, where precise energy determinations are crucial.

Frequency and Energy Relationship

In the realm of NMR and quantum physics, there is a direct relationship between frequency and energy. This can be seen clearly in the equation \( E = hu \), where \( E \) is energy, \( h \) is Planck’s constant, and \( u \) is the frequency.
Essentially, the higher the frequency of the NMR spectrometer, the more energy is required for each proton spin-flip.

• Frequency acts like a lever; higher frequencies imply higher energies.
• Increasing the NMR frequency from 200 MHz to 300 MHz increases the energy required.

This principle is fundamental in determining the energy landscape of atomic nuclei. It helps in interpreting the spectroscopic data, thereby enhancing our understanding of molecular structures based on how they interact with magnetic fields.