X-ray diffractometers often use metals that have had their core electrons excited as a source of X-rays. Consider the 2p → 1s transition for copper, which is called the K⍺ transition. Calculate the wavelength of X-rays (in Å) given off by the K⍺ transition if the energy given off by a mole of copper atoms is 7.77⨉10⁵ kJ.(1Å = 10^-10 m)

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hey everyone in this example, we're told that the energy given off by a mole of iron atoms by an alpha transition is 6.15 times 10 to the fifth power kilo jules. We need to calculate the wavelength of our x rays in extremes given off by our alpha transition. So what we want to do is recognize that our energy is given in units of kilo jules and we want to convert this in units of jewels. So we're going to take this value 6.15 times 10 to the fifth power kayla jewels. And this is per photon of light. And we're going to first multiply by our first conversion factor to cancel out that unit photon. So we should recall that for one mole of photons That corresponds to avocados number, which we recall would be 6.02, 2 times 10 to the 23rd power photons. So now we're able to cancel out our units of photons and now we want to multiply by our next conversion factor. Where we're going to recall that our prefix kilo jewel Tells us that we have 10 to the third power jewels. So we have one kg jewel. And what we're going to do is be able to cancel out our units of kilo jewels, leaving us with jewels as our unit for energy. And what we're going to get is a value Equal to 1.02, 1 times 10 to the negative 15th power jewels. And so according to the prompt, we want to solve for our wavelength. So we're going to recall that wavelength is represented by the symbol lambda. And we can calculate it by taking plank's constant, multiplied by the speed of light and dividing by our value for energy of a photon. And so now we're going to plug in what we know. So we should recall that plank's constant is equal to 6.626 times 10 to the negative 34th power in units of joules times seconds. We're going to recall that our speed of light is going to be three point oh times 10 to the eighth power units of meters per second. And then in our denominator we're going to plug in that energy value that we just converted Which was equal to 1.02, 1 times 10 to the negative 15th power in units of jewels as we found above. And so canceling out our units, we can get rid of the value jewels, we can get rid of seconds with inverse seconds. And this leaves us with units of meters as our final unit for wavelength. And what we should get here is a value of 1.95 times 10 to the negative 10th power in units of meters. And because according to the prompt, we want our final answer for wavelength to be in units of extremes. As the prompt stated, we're going to recall that for one m, We have a total of 10 to the 10th power angstrom. And so we're going to use this as a conversion factor. So we're going to take our 1.95 times 10 to the negative 10th power meters and we're going to divide by one m For 10 to the 10th power and streams. Now we're able to cancel out our units of meters, leaving us with angstrom as our final unit for wavelength. And what we would get Is that our wavelength is equal to 1.95 Angstrom. This is our wavelength of our X rays. So to complete this example, we have what's highlighted in yellow as our final answer. So I hope that everything I explained was clear. If you have any questions, please leave them down below and I will see everyone in the next practice video.

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Key Concepts

X-ray Emission

Energy-Wavelength Relationship

Mole Concept and Energy Conversion