According to the equation for the Balmer line spectrum of hydroge...

Question

According to the equation for the Balmer line spectrum of hydrogen, a value of n = 3 gives a red spectral line at 656.3 nm, a value of n = 4 gives a green line at 486.1 nm, and a value of n = 5 gives a blue line at 434.0 nm. Calculate the energy in kilojoules per mole of the radiation corresponding to each of these spectral lines.

Accepted Answer

Hi everyone. This problem reads the most distinct lines in the atomic spectra of helium R 447.1 nanometers. 501.6 nanometers and 587.6 nanometers. What is the energy? Energy and killing joules per mole of each of these spectral lines. Okay, so the question that we want to answer is the energy and killing joules per mole of each of these spectral lines. So we have three spectral lines that we're going to want to calculate the energy for. And the equation that we're going to want to use that shows the relationship between the energy of a photon and the wavelength of light is energy is equal to plank's constant times the speed of light over lambda. Okay. And planks constant is represented by six point 6, 26 times 10 To the -30 for Jules Times seconds. And the speed of light is represented by 3.0 times 10 to the eighth meters per second. And lambda, our wavelength, because our speed of light is in meters. We want lambda and meters. Okay. And so here we see that for our three spectral lines, the wavelength is given in nanometers. So we're going to want to convert these wavelengths from nanometers two m. So let's go ahead and start off by calculating the energy for our first spectral lines. So we're going to do the energy for the blue line. Okay, so for the blue line, let's start off by converting our wavelength. Okay, so for our blue line we're told the wavelength is 447.1 nanometers. So we want to go from nanometers to meters. Okay. And one nanometer, there is 10 to the - m. So our units for nanometers cancel and we're left with meters. So when we do this calculation we get 4. times 10 to the negative seven m. Okay, so let's go ahead and calculate the energy now. So we know energy is equal to Planck's constant 6.626 times 10 to the negative 4 34. Excuse me, jules times seconds. Okay, times the speed of light. And this is all over lambda. Our wavelength. So we're gonna put the wavelength in meters that we just calculated. So 4.471 times 10 to the negative seven m. Alright, so the energy that we're going to get when we do this calculation is our energy is equal to, we'll put it here. Our energy is equal to 4. 59, eight, five Times 10 to the - jewels. And this is going to be jewels per photon. So that's the energy per photon. But the question asked us. And for the energy and kill a jules per mole. So we need to go from jules per photon to kill a jules per mole. So let's start off by going from photons to moles and we'll do this using avocados number. Okay, so in one mole Of anything avocados number tells us we have 6.02, 2 times 10 to the 23. Okay. And in this case photons. All right. So our units for photons cancel. And now we have jewels per mole. But we want killer jewels per mole. So we need to go from killer jewels or we need to go from jewels to kill a jules. Okay. And one killer jewel, There is 1000 jewels. So, our units of jewels cancel. And now we have killer joules per mole, which is what we want. All right, So let's go ahead and do this calculation. So, our energy and killing joules per mole for our blue line is going to equal 267.7 kg jewels per mole. And well, right here that this is for our blue line. All right. So, we just calculated one out of three. So, let's go ahead and calculate for our cyan line. Alright, We're going to do the same thing. All right, so we're going to do the same thing, but for our cyan line. So we need to first convert our wavelength in nanometers two m. So, our wavelength for our cyan line and nanometers is 501.6 nanometers. And we want to go from nanometers two m. Okay. In one nanometer there is 10 to the negative nine m nanometers cancel and we're left with meters. So when we do this calculation, we're going to get our wave like this equal to 5.16 times 10 to the negative seven m. Alright, so now we can calculate the energy. Okay, our energy is going to equal Plank's constant 6.62, 6 times 10 to the - jules times seconds, times our speed of light, 3.0 times 10 to the eight m per second. And this is all over wavelength in meters. So we just calculated our wavelength for our cyan line. It is 5.16 times 10 to the negative seven m. Okay, so let's go ahead and right out our energy. So our energy for a cyan light is going to equal 3. times 10 To the negative 19 jewels per photon. Now we want to go from Jules per photon to kill jules per mole. So let's first go ahead and go from photons to moles. And one more. There is 6.02, 2 times 10 to the photons. Okay, so we have photons canceled and now we have jewels per mole but we want killer joules per mole. Okay, so in one kill a jewel There is 1000 jewels. Our units for jewels cancel. And now we have killer joules per mole. Alright, so let's go ahead and do our calculation. And when we do our calculation we're going to get the energy is equal to 238.6 kg jewels per mole. And this is the energy for the cyan line. Alright, so now we know the energy for our second spectral line and we have one more spectral line to go. And our last spectral line is the yellow line. Alright, so let's go ahead and calculate the energy for the yellow line. Alright, so we need to convert the wave length for a yellow line from nanometers two m. So the wavelength is 587.6 nanometers. And we need to go from nanometers two m in one nanometer, there is 10 to the negative nine m. Units of nanometers cancel. And we're left with units of meters. So when we do this calculation we're going to get 5. Six times 10 to the -7 m. So now we can calculate the energy, the energy is going to equal, our energy is going to equal plank's constant 6.626 times 10 to the negative 30 for jules, times seconds, times the speed of light, three m, three times 10 to the eight m per second. And this is all over the wavelength in meters. And we just calculate that the wavelength is 5.876 times to the negative seven m. Okay, so the energy When we do the calculation is going to equal 3.38- times 10 to the -19 jewels per photon. And we want to go from jules per photon to kill a jewels per mole. Okay. And one mole, There is 6.022 times 10 to the 23 photons. Okay, so our units of photons cancel and now we have units of jewels per mole. And we want killer jewels promote and one killer jewel, There is 1000 jewels. Our units of jewels cancel. And we're left with killer joules per mole. So we can go ahead and write that the energy is going to equal 203. kg joules per mole for our yellow line. Okay, so that is going to be the final answer. So we just calculated the energies and the energy and jewels per mole of each of these spectral lines. Alright, so for our blue line, our cyan line and our yellow line, that is it. For this problem, I hope this was helpful

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