Problem 2 - Chemistry: Principles and Reactions

Question

Magnetic resonance imaging (MRI) is a powerful diagnostic tool used in medicine. The imagers used in hospitals operate at a frequency of \(4.00 \times 10^{2} \mathrm{MHz}\left(1 \mathrm{MHz}=10^{6} \mathrm{~Hz}\right) .\) Calculate (a) the wavelength. (b) the energy in joules per photon. (c) the energy in kilojoules per mole.

Step-by-Step Solution

Answer

Question: An MRI machine operates at a frequency of \(4.00 \times 10^2 MHz\). Calculate the following: a) The wavelength of the radiation b) The energy in joules per photon c) The energy in kilojoules per mole Answer: a) Wavelength: 7.5 x \(10^{-2}\) m b) Energy per photon: 2.65 x \(10^{-25}\) J c) Energy per mole: 0.1595 kJ/mol

Step 1: Convert the frequency to Hz

The frequency given is in MHz, so we need to convert it to Hz by multiplying with \(10^6\). This will give us: \(f = 4.00 \times 10^2 MHz \times 10^6 Hz/MHz = 4.00 \times 10^8 Hz\)

Step 2: Calculate the wavelength

Using the speed of light relation \(c = f * \lambda\), we can find the wavelength as follows: \(\lambda = \frac{c}{f} = \frac{3.00 \times 10^8 m/s}{4.00 \times 10^8 Hz} = 0.750 \times 10^{-1} m = 7.5 \times 10^{-2} m\)

Step 3: Calculate the energy per photon

Using the energy per photon formula \(E = h * f\), we can calculate the energy per photon: \(E = (6.626 \times 10^{-34} Js) \times (4.00 \times 10^8 Hz) = 2.65 \times 10^{-25} J\)

Step 4: Calculate the energy per mole

Finally, using the energy per mole formula \(E_{mole} = E * N_A\), we can calculate the energy per mole: \(E_{mole} = (2.65 \times 10^{-25} J) \times (6.022 \times 10^{23} mol^{-1}) = 0.001595 \times 10^2 kJ/mol = 0.1595 kJ/mol\) So, the answers for each part of the question are: a) Wavelength: 7.5 x 10^{-2} m b) Energy per photon: 2.65 x 10^{-25} J c) Energy per mole: 0.1595 kJ/mol

Key Concepts

Understanding Wavelength Calculation

When it comes to calculating wavelength in the context of magnetic resonance imaging (MRI), we start with the frequency provided. MRI imagers use radiofrequency waves at noticeable frequencies, usually expressed in megahertz (MHz). Fortunately, converting MHz to a more manageable unit like hertz (Hz) is simpler than it may initially seem. You multiply the frequency in MHz by \(10^6\) to switch over to Hz.

Once in Hertz, you can find the wavelength \(\lambda\) by using the formula for the speed of light: \(c = f \times \lambda\), where \(c\) is the speed of light (approximately \(3.00 \times 10^8\) meters per second). From this equation, the wavelength expression becomes \(\lambda = \frac{c}{f}\).

Calculating and substituting the values for an MRI frequency gives us the wavelength of the radio waves used in these diagnostic machines. Understanding this step effectively connects concepts of electromagnetic theory with practical applications in medical imaging.

Understanding Photon Energy

Photon energy is a crucial concept, especially when dealing with applications like Magnetic Resonance Imaging (MRI). Energy of a single photon can be calculated using the equation \(E = h \times f\). Here, \(E\) represents the energy of a photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34} Js\)), and \(f\) is the frequency of the electromagnetic wave.

By substituting the known values, you can determine the energy possessed by an individual photon at a given frequency, allowing insights into how MRI uses electromagnetic waves for imaging. While these calculations show that photon energy is minuscule individually, it is the collective action of many such photons that enables MRI to function so effectively in creating images.

The Importance of Frequency Conversion in MRI

Converting frequency to the appropriate unit is a foundational step in many calculations, especially when dealing with electromagnetic waves in medical technology. For MRI, knowing the frequency of the operating system in megahertz is the starting point. These figures must be converted to hertz for calculations, since Hertz is the standard SI unit for frequency.

This conversion ensures consistency across all calculations involving electromagnetic waves, particularly those related to wavelength and photon energy. Making these conversions successfully in the beginning paves the way to accurately understanding the interactions and behavior of waves in systems like MRI. It aligns the parameters into compatible units, clarifying the rest of the mathematical steps involved.

Spectroscopy Education: A Gateway to Understanding MRI

Spectroscopy is a field that deals with the study of how matter interacts with electromagnetic radiation. It lays down the scientific foundation for understanding MRI, which can be seen as applied spectroscopy in a medical setting. At its core, spectroscopy education teaches about the different types of electromagnetic waves, including their frequencies and wavelengths.

Studying spectroscopy gives students a comprehensive insight into how MRI works, taking advantage of radio frequencies to create detailed images of body tissues. This understanding enables students to better appreciate not only the technology but also the physics principles at play, such as energy calculations and wave behaviors.

• Learning about different spectroscopic techniques can help in diversifying knowledge that applies to many fields in physics and chemistry.
• Fundamentals learned in spectroscopy provide the groundwork necessary for advanced imaging techniques used in medicine.

Embracing this education ensures a solid comprehension of MRI technology and its principles, enhancing both practical and theoretical knowledge.