91Ó°ÊÓ

In the world of physics, the energy of a photon is a crucial concept. A photon is a tiny packet of light energy, and to determine its energy, we rely on Planck's equation: \( E = h \cdot u \), where:

• \( E \) is the energy of the photon.
• \( h \) stands for Planck's constant, which equals \( 6.626 \times 10^{-34} \text{ J s} \).
• \( u \) is the frequency of the photon.

In our exercise, the frequency \( u \) is \( 400 \times 10^6 \text{ Hz} \). By substituting these values into the equation, we calculate the energy of one photon to be \( 2.6504 \times 10^{-25} \text{ J} \). This precise measure of photon energy is foundational in understanding how electromagnetic waves like those used in MRI machines work.

Avogadro's number is a very large constant used in chemistry and physics, where it defines the number of entities, usually atoms or molecules, in one mole of a substance. This constant is approximately \( 6.022 \times 10^{23} \text{ mol}^{-1} \).
This number helps to bridge the microscopic world of atoms to the macroscopic world we observe. In the context of our exercise, once we determine the energy of a single photon, we multiply this by Avogadro's number to find the total energy contained in a mole of photons. Thus, \( E_{\text{mole}} = 2.6504 \times 10^{-25} \text{ J} \times 6.022 \times 10^{23} = 1.596 \times 10^{-1} \text{ J/mol} \).
By using Avogadro's number, we can scale up the energy from a single photon to a quantity like a mole, providing insight into energy calculations on a bulk level.

The frequency conversion is an essential step when dealing with electromagnetic radiation. Frequency (\( u \)) represents how many cycles of a wave pass a point in one second and is measured in Hertz (Hz).

• In our specific scenario, we started with \( 400 \text{ MHz} \), which is equivalent to \( 400 \times 10^6 \text{ Hz} \).
• The process basically involves multiplying by the factor of \( 10^6 \) to convert from megahertz (MHz) to hertz (Hz).

Understanding frequency conversion is crucial when you want to ensure real-time calculations align with standard units, ensuring consistency, especially in scientific contexts like calculating photon energy. This precision allows us to employ Planck's equation effectively.

When dealing with energy in chemistry and physics, converting joules to kilojoules per mole is common practice. The mole is a standard unit in chemistry, and energies are often expressed in this way for convenience and meaningful comparison.

• In the given exercise, once we've calculated the energy in joules per mole (\( 1.596 \times 10^{-1} \text{ J/mol} \)), we convert this figure into kilojoules by dividing by 1000, as there are 1000 joules in a kilojoule.
• This conversion yields \( 1.596 \times 10^{-4} \text{ kJ/mol} \), which is far more manageable and useful for chemical and physical calculations.

By converting to kilojoules per mole, we align the energy calculations to a convention frequently used in practical applications like energy transfer and reaction dynamics.