Ultraviolet light with wavelength 75 nm shines on hydrogen atoms in their ground states, ionizing some of the atoms. What's the energy of the electrons freed in this process?

Ultraviolet (UV) light is a form of electromagnetic radiation that is invisible to the human eye and has a wavelength shorter than that of visible light but longer than X-rays. UV light’s wavelength ranges from about 10 nm to 400 nm and is beyond the violet section of the spectrum of visible light, hence 'ultraviolet'.

When UV light hits certain materials, it can cause electrons to be ejected, which relates to the photoelectric effect. In our exercise, a UV light with a wavelength of 75 nm shines on hydrogen atoms. The energy of UV light is high enough to cause ionization of the hydrogen atoms, which results in the liberation of electrons.

Scientists use wavelengths like UV light to study the properties of atoms. For instance, by observing the effect of UV light on hydrogen atoms, the energy escape of electrons can be calculated. This provides essential information in fields such as quantum mechanics and also in practical applications such as in the development of solar panels.

Planck's constant (\( h \) is a fundamental physical constant that is vital to the field of quantum mechanics. It denotes the quantization of energy levels in particles and relates energy to the frequency of radiation.

\( h \) has the value of approximately 6.626 x 10^{-34} joule-second (Js) and is central to the equations that describe the behavior of photons and atoms. In our exercise, Planck's constant is used to bridge the gap between the frequency of the UV light used and the energy of the released electrons.

The formula \( E = hu \) illustrates this relationship, where \( E \) represents the energy of a photon and \( u \) is the frequency of the electromagnetic radiation. With the knowledge of Planck's constant and the frequency of the light, one can calculate the energy of the photons that interact with the hydrogen atoms.

The photoelectric effect is a phenomenon where electrons are ejected from the surface of a material after exposure to light or electromagnetic radiation. This groundbreaking concept was explained by Albert Einstein, confirming the particle nature of light, for which he was awarded the Nobel Prize in Physics.

The photoelectric effect demonstrates that light can be thought of as consisting of particles, or photons, and when they have sufficient energy, they can release electrons from a material. The energy of each photon must be above a certain threshold—that of the material's work function—for the ejection of electrons to occur.

In our exercise involving hydrogen, the photoelectric effect is at play when UV light ionizes the atoms. Since the UV light's wavelength corresponds with high-frequency, high-energy photons, these photons have enough energy to overcome the ionization energy of hydrogen and release an electron. The energy of the freed electrons is then the energy of the incoming photons minus the ionization energy of the hydrogen atoms. Understanding the photoelectric effect not only helps in comprehending exercises like this but also lays the foundation for modern electronics, including the development of photovoltaic cells and photoelectron spectroscopy.