Question

The following \({ }^{1} \mathrm{H}\) NMR absorptions were obtained on a spectrometer operating at \(300 \mathrm{MHz}\). Convert the chemical shifts from \(\delta\) units to hertz downfield from TMS. (a) \(2.1 \delta\) (b) \(3.45 \delta\) (c) \(6.30 \delta\) (d) \(7.70 \delta\)

Short answer

(a) 630 Hz, (b) 1035 Hz, (c) 1890 Hz, (d) 2310 Hz.

Step-by-step solution

Understand the Concept of Chemical Shift

Chemical shift in NMR spectroscopy is expressed in parts per million (ppm, denoted as \( \delta \)), which is a dimensionless quantity representing a fraction of the spectrometer's operating frequency.

Formula for Conversion

The chemical shift \( \delta \) in ppm is related to the spectrometer frequency (\( u_0 \)) in megahertz (MHz) by the formula: \[ \text{Hz} = \delta \times u_0 \] Where \( u_0 \) is the operating frequency of the NMR spectrometer in hertz.

Convert 2.1 \(\delta\) to Hz

Given \( \delta = 2.1 \) ppm and \( u_0 = 300 \) MHz, convert to hertz: \[ \text{Hz} = 2.1 \times 300 \times 10^6 = 630 \, \text{Hz} \]

Convert 3.45 \(\delta\) to Hz

Given \( \delta = 3.45 \) ppm, convert to hertz as follows: \[ \text{Hz} = 3.45 \times 300 \times 10^6 = 1035 \, \text{Hz} \]

Convert 6.30 \(\delta\) to Hz

Given \( \delta = 6.30 \) ppm, convert to hertz: \[ \text{Hz} = 6.30 \times 300 \times 10^6 = 1890 \, \text{Hz} \]

Convert 7.70 \(\delta\) to Hz

Given \( \delta = 7.70 \) ppm, convert to hertz: \[ \text{Hz} = 7.70 \times 300 \times 10^6 = 2310 \, \text{Hz} \]

Key concepts

Chemical Shift

Chemical shift is a crucial concept in NMR spectroscopy, serving as a way to differentiate between various environments of atomic nuclei within a molecule. When you hear about the "chemical shift," it refers to the resonance frequency of a nucleus relative to a standard in a magnetic field.
Generally, tetramethylsilane (TMS) is used as the reference compound. The chemical shift is measured in parts per million (ppm) and labeled as \( \delta \). This value captures the influence of surrounding atoms and bonds on nuclear magnetic activity.
By reflecting these environmental effects, chemical shifts offer insights into the molecular structure. The more shielded a nucleus, the lower its chemical shift appears, and vice versa. This means that if electron density is high around a nucleus, it experiences less of the applied magnetic field, making its deviation from TMS smaller.
Utilizing chemical shifts, chemists can determine functional groups and structural details of molecules. For example, protons attached to a carbon next to an electronegative group will often appear downfield (higher \( \delta \)) than those next to a more electropositive environment.

Parts Per Million (ppm)

Parts per million (ppm) is a unit of measurement often used to express chemical shifts in NMR spectroscopy. Basically, ppm is a dimensionless quantity that represents how far the resonance frequency of a nucleus is compared to the reference (usually TMS).
Here's why it's useful: by using ppm, we can express chemical shifts independently of the spectrometer's frequency. This way, a chemical shift measured in ppm on a 300 MHz spectrometer would be identical on a 600 MHz spectrometer, making it a universal measurement.
To convert from ppm to hertz (Hz), you need to know the spectrometer frequency represented by \( u_0 \). The conversion formula is:

• \( \text{Hz} = \delta \times u_0 \)

Where \( \delta \) is the shift in ppm and \( u_0 \) is the frequency in hertz. This formula allows chemists to easily calculate the frequency shift from the reference point in absolute terms.

Spectrometer Frequency

Spectrometer frequency is an essential factor in NMR spectroscopy, fundamentally influencing the measurement of chemical shifts. It is expressed in megahertz (MHz) and ultimately determines the resolution and quality of the spectrum obtained.
Different spectrometers operate at varying frequencies, typically ranging from 60 MHz to over 1 GHz for protons. The frequency of your spectrometer defines the strength of the magnetic field being applied and hence, the energy levels between which the nuclei resonate. Higher frequencies allow for better resolution, giving sharper, more defined peaks in the spectra.
In the context of chemical shift conversion, the spectrometer's frequency is integral for calculating the shift in hertz. For example, with a spectrometer operating at 300 MHz, the chemical shifts for different \( \delta \) values can be transformed into hertz using the formula:

• \( \text{Hz} = \delta \times \text{frequency in MHz} \times 10^6 \)

Understanding your spectrometer's frequency isn't just about conversions—it directly impacts how detailed and precise your analysis will be.