The conversion of violet radiation (800 nm) into x-ray radiation (32nm) was reported in 1998. What is the minimum number of violet photons needed to form one of the x- ray photons? A) 4. B)16. C)25. D) 40
We simply divide the longer wavelength by the shorter wavelength: `800/32 = 25`
However, if you want to know the energies, and why the above solution is justified, we can first solve for the energies:
The wavelength (related to the color of the radiation) corresponds to certain energies. These energies are quantized. Hence, a photon of a certain energy would have a certain wavelength, and associated energy. We note that the quantization of energy is related to Planck's constant:
`E = nhnu`
n is the number of photons
`nu` is the frequency in hertz (or `1/s` ), and `h = 6.62 times 10^(-34) (m^2 kg)/s`
The wavelength and frequency are related:
`c = lambda nu`
where `c` is the speed of light is `3 times 10^8 m/s^2` .
We need to calculate the energy associated with the violet radiation and the x-ray. We first convert the wavelengths to meter.
`800nm = 8 times 10^(-7) m` for the violet radiation.
`32nm = 3.2 times 10^(-8)m` for the x-ray.
We then calculate the frequencies:
`nu = c/lambda`
`nu_(p) = (3 times 10^8)/(8 times 10^(-7)) = 3.75 times 10^14 Hz`
`nu_(x) = (3 times 10^8)/(3.2 times 10^(-8)) = 9.38 times 10^14`
We calculate the associated energy:
`E_p = (6.62 times 10^-34)*(3.75 times 10^(14)) = 2.48 times 10^-19`
`E_x = (6.62 times 10^-34)*(9.38 times 10^14) = 6.21 times 10^-18`
Here, we assumed that we only have one photon of each. Now, we want to know how many violet photons would have the same energy as one x-ray:
`nhnu_p = E_x`
But, we already know that the calculate energy is for a single photon, `E_p = hnu_p`
`n = (E_x)/(E_p)= (6.21 times 10^-18)/(2.48 times 10^-19) = 25`
Thus, we need 25 of the photons, letter C.
The energy of any form of electromagnetic radiat is directly proportional to the frequency and inversely proportional to the wavelength of the EMR. Energy - Planck's constant x frequency OR Energy = Planck's Constant x speed of light / wavelength Violet light has a wavelength of 800nm = 8.00X 10-7 meters X-ray radiation has a wavelength of 32nm = 3.2 x 10-8 meters Planck's constant = 6.634 X10-34 Speed of light = 3 x 10+8 m/s Violet light = (6.634x10-34 * 3 x 10+8) /8.00x 10-7 = 2.488x10-19j X-ray = (6.634x10-34 ^ 3 X10+8) / 3.2 x10-8 = 6.221 x 10-18j Energy of X-rays / Energy of Violet light = 25 times greater value