In which region of the electromagnetic spectrum do we find the energy to fully ionize hydrogen?
A) Radio
B) Infrared
C) Visible
D) Ultra-violet
E) X-ray
I want to know how you would go about doing this question; I know we need to find the wavelength, but what exactly is the connection between wavelength and energy required?
Energy= boltzmanns constant*speedlight/wavelength
Energy= inonization energy
wavelength= incoming light
solve for wavelength.
Remember: MEMORIZE
Planck's equation
energy= boltzmannconstant*frequency
E)X-Ray region
To determine the region of the electromagnetic spectrum where we find the energy required to fully ionize hydrogen, we need to understand the relationship between wavelength and energy.
First, we know that the energy of a photon of light is directly proportional to its frequency (E ∝ ν) and inversely proportional to its wavelength (E ∝ 1/λ). The constant of proportionality is Planck's constant, denoted as h (E = hν or E = hc/λ, where c is the speed of light).
Now, let's consider ionizing hydrogen. Fully ionizing hydrogen means removing its only electron. The energy required to remove the electron from the ground state of a hydrogen atom is referred to as the ionization energy, which is approximately 13.6 electron volts (eV).
Since we need to find the region of the electromagnetic spectrum that provides enough energy to fully ionize hydrogen, we can use the formula E = hc/λ to calculate the energy required for different wavelengths.
To determine the wavelength associated with the required energy of 13.6 eV, we can convert the given energy into joules and plug it into the equation to solve for λ.
First, we convert 13.6 eV to joules: 1 eV = 1.6 x 10^-19 Joules, so 13.6 eV = 13.6 x 1.6 x 10^-19 J = 2.18 x 10^-18 J.
Using the equation E = hc/λ, we plug in the value for energy (2.18 x 10^-18 J) and solve for λ:
2.18 x 10^-18 J = (6.63 x 10^-34 J·s)(3 x 10^8 m/s) / λ
Rearranging the equation to solve for λ:
λ = (6.63 x 10^-34 J·s)(3 x 10^8 m/s) / (2.18 x 10^-18 J)
λ ≈ 9.09 x 10^-8 m (or 90.9 nm)
Now that we have the wavelength, we can compare it to the regions of the electromagnetic spectrum listed in the options:
A) Radio: Radio waves have wavelengths ranging from meters to kilometers, which is much longer than 90.9 nm. Hence, radio waves do not have sufficient energy to fully ionize hydrogen.
B) Infrared: Infrared waves have wavelengths ranging from a few hundred nanometers to several micrometers, which is longer than 90.9 nm. Therefore, infrared waves do not have enough energy to fully ionize hydrogen either.
C) Visible: Visible light waves have wavelengths ranging from 380 nm (violet) to 750 nm (red), which is within the range of the calculated wavelength (90.9 nm). Thus, visible light does have enough energy to fully ionize hydrogen.
D) Ultra-violet: Ultraviolet waves have wavelengths shorter than visible light, typically ranging from 10 nm to 400 nm. Since the calculated wavelength (90.9 nm) falls within the range of ultraviolet waves, we can conclude that ultraviolet waves provide enough energy to fully ionize hydrogen.
E) X-ray: X-rays have even shorter wavelengths (less than 10 nm) than the calculated wavelength (90.9 nm). Therefore, x-rays also have enough energy to fully ionize hydrogen.
In conclusion, the region of the electromagnetic spectrum where we find the energy required to fully ionize hydrogen is in the ultraviolet and x-ray regions (options D and E).