# 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.  ...

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

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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`

Hence:

`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.

Sources:

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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