(b) Calculate
the energy of a photon of radiation whose waveleng...

Question

(b) Calculate 
the energy of a photon of radiation whose wavelength is 413 
nm.

Accepted Answer

everyone in this example. We need to find the energy for each electromagnetic radiation expressing our answer in joules per moles. So before we go ahead and start part A, we're going to recall our formula for energy. And we should recall that energy is equal to Planck's constant times our frequency if frequency is given but were not given frequency were given wavelength. So we want to recall our relationship where the speed of light is equal to r wave length, multiplied by frequency. And so we can go ahead and simplify this so that we can solve for frequency by taking our speed of light and dividing it by our wavelength. And so we can go ahead and substitute this equation into our energy formula. So what we would have is ultimately therefore energy is equal to Planck's constant times the speed of light divided by our given wavelength. So let's go ahead and start part A. For part A. We would have energy is equal to Planck's constant, which we should recall is 6.626 times 10 to the negative. 34th power jewels times seconds. Then we're going to be multiplying by the speed of light which we recall is three times 10 to the eighth power meters per second. And then in our denominator we're gonna plug in our given wavelength. So we're given a wavelength of .149 nm in part a. But we want to recall that wavelength which is our lambda symbol here is in units of meters. So we're gonna convert from nanometers into meters and we should recall that our prefix nano gives us 10 to the ninth power meters. This allows us to cancel out our units of nanometers, leaving us with meters and then we can go ahead and make this simpler by getting rid of our other units so we can get rid of meters and we can also get rid of seconds. And so we're just left with jewels for energy, which is what we want. And so what we're going to get is that our energy is equal to in our numerator we should have when we take the product of our numerator, we're going to get 1.987, 8 times 10 to the -25th power. And then in our denominator we're going to have our converted wavelength two m as 1.49 times 10 to the negative 10th power. And so now we can take the value of this quotient here And we still have jewels in our numerator. Sorry about that. So this is going to be equal to a value of 1.33 times 10 to the negative 15th power jewels as our energy. And again, the question wants us to give our energy and kilo joules per mole. So we're going to make a conversion where we're going to go from jewels, two kg joules per mole. So what we're gonna do is take our energy set that equal to a value of What we just found above, which is 1.33 times 10 to the negative 15 power jewels per photon of light or per x ray photon. So this is going to be then multiplied by our conversion factor to go from jules and tequila jules. And so we should recall that our prefix kilo gives us 10 to the third power or 10 to the third power jewels as our denominator. And so we can go ahead and get rid of our jewel unit. So now we have kayla jewels and now we just need to end up with moles. And we're gonna multiply this by the next conversion factor that we should recall, Which is that we have 6.022 times 10 to the 23rd power photons Permal. And so now we have photons that we can cancel out leaving us with kayla jewels per mole. Which is what we want to answer this question. And so this is going to give us a value equal to eight, 800926. And to write this in scientific notation, we'll just say 8.01 times 10 to the negative 5th power kilo, joules per mole as our energy for an x ray photon at the given wavelength. So this is our answer for part A. And now we can go ahead and move on to part B. So we're going to follow the same process for part B. Sorry about that. So what we're going to have is our energy is equal to six 0.626 times 10 to the negative 34th power jewels times seconds. And again we want to keep this and multiply this by the speed of light which is three times 10 to the eighth power meters per second. So in our denominator we're going to go ahead and above plug in that given wavelength of our, of our gamma ray photon. And so in our denominator we're going to plug in to sorry 2.45 times 10 to the negative fifth power nanometers. And we want to again convert to the proper units for our wavelength into meters. So we're going to go from nanometers in the dominator two m. So we recall that our prefix nano gives us 10 to the ninth power meters. And we can go ahead and cancel out nanometers leaving us with meters and this is going to make it easier to cancel out the rest of our units so we can go ahead and get rid of the meters as well and then seconds leaving us with jewels in our numerator. So our next step is to simplify our numerator so that we have energy is equal to The product of the two parentheses in our numerator which gives us 1.987, 8 times 10 to the night of 25th power. And then we have just jewels in our numerator In our denominator we still have 2. or actually we just converted into meters. And so our calculation and our denominator is 2.45 times 10 to the -14 meters for a wavelength. And so now we're going to divide our numerator by our wavelength and our denominator. And this is going to give us our energy in jewels equal to 8. comes 10 to the negative 12 powered jewels. Now again, we want to finish off our final answer in kilo joules per mole. So we're going to follow that same conversion factor as before, Where we took our energy value of 8.11 times 10 to negative power jewels which was our energy per photon of our Gamla race. Sorry, that should be an end. And this is going to be then multiplied by our conversion factor to go from jules, tequila jewels. And we should recall that our prefix kilo Gives us 10 to the third power jewels For one kg jewel. So we are allowed to now cancel out our jewels. Now we just have kilograms per photon. And what we really want is kilo joules per mole. So we're going to multiply by the conversion factor for a photon where we have in our numerator because we want to cancel out photons 6.22 times 10 to the 23rd Power photons Permal. And so this allows us to cancel out photons leaving us with kayla jules Permal as our final units. So what we're going to get is a value equal to 4.88 times 10 to the 9th power kilo joules per mole. And so this would be our final answer for Part B, given our wavelength for our gamma ray photon in Part B. So everything boxed in blue. Our final answers. If you have any questions, please leave them down below and I will see everyone in the next practice video.

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