Pure silicon at room temperature contains approximately 1.0 × ...

Question

Pure silicon at room temperature contains approximately 1.0 × 10¹⁶ free electrons per cubic meter. (a) Referring to Table 25.1, calculate the mean free time t for silicon at room temperature. (b) Your answer in part (a) is much greater than the mean free time for copper given in Example 25.11. Why, then, does pure silicon have such a high resistivity compared to copper?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A pure ger geranium crystal as a resistivity of 0.60 ohms multiplied by meters and 2.5 multiplied by 10 to the power of 13 electrons per cubic centimeters at 20 °C. A determine the mean free time for geranium at this temperature B determine the mean free time for geranium when it is greater than the mean free time for some conducting materials. Provide a reason why there is a high resistivity for geranium relative to a conductor. OK. So that's our end goal. So our end goal is to find two separate answers. We need to find an answer for part A and for part B for A, we're trying to determine the mean free time for geranium at 20 °C. And then B we're trying to determine the mean free time for geranium when it's greater than the mean free time for some conducting materials. And then we also need to provide a reason for why there's a high rec civility for geranium relative to a conductor. OK. So we're given some multiple choice answers. So all the answers for part A are in seconds and then we're also given for part like answers for part B which is a little reason why. OK. So let's read off our multiple choice answers to see what our final answer pair might be. A is for part A 2.33 multiplied by 10 to the power of negative 12. And B is resistivity is inversely proportional to the mean free time. B is for part A 2.33 multiplied by 10 to the power of negative 20. And B is geranium has a low concentration of charge carriers. C is for part A is 2.37 multiplied by 10 to the power negative 20. And B is resistivity is inversely proportional to mean free time. And D is for part A 2.37 multiplied by 10 to the power of negative 12. And B is geranium has a low concentration of charge carriers. OK. So to solve for part A first off, let us recall and use the equation for resistivity which states that row is equal to M divided by N multiplied by E squared multiplied by Tau. OK. And let's call this equation one. So M is the electron mass and the, for the letter M as in mom and is the electron mass. And then N, the letter N as in not or knots with a K huh but N is equal to the current carrier and E is the electron charge magnitude. And tau is the mean time between collisions. OK. So now we need to quickly convert, before we go any further, we need to quickly convert electrons per centimeters cubed to electrons per meters cubed. So let's do that. So N which as we know is the current carrier is equal to and this is provided to us in the pro as 2.5 multiplied by 10 to the power of 13. And I'm just gonna abbreviate it as el for electrons per centimeters cubed. So using a little bit of dimension analysis here, we need to multiply it by so there's 100 centimeters in 1 m and we need to cube this value. So 100 centimeters divided by 1 m, all cubed is equal to 2.5 multiplied by 10 to the power of 19 electrons per meters. Cute fantastic. So now we can dive right on in. So now we need to solve for Tau. So we need to rearrange equation one to solve for Tau. So when we do that, we get that Tau is equal to M divided by N multiplied by E squared multiplied by row. OK. So now we can plug in all of our known variables to solve for Tau. So let's do that. So Tau is equal to 9.11. So this is the mass of an electron. So 9.11 multiplied by 10 to the power of negative 31 kg divided by N which is 2.5 multiplied by 10 to the power of 19 electrons per meters cubed multiplied by E. And the value for E is 1.602 multiplied by 10 to the power of negative 19 electrons square multiplied by row which the value for row, which is the resistivity is 0.60. And this is provided to us in the problem. Ohms multiplied by meters. Awesome. So when we plug that into a calculator, we get that Tau is equal to 2.366 multiplied by 10 to the power of negative 12 seconds. But as we noticed in our multiple choice answers, it's to two decimal places. So let's round it up. So when we round it, we will get 2.37 multiplied by 10 to the power of negative 12 seconds. And that is our final. And for Tau and this is also part a Awesome. So now we have found the mean free time for geranium at 20 °C and it's 2.37 multiplied by 10 to the power of negative 12. And also really fast, I'll write it in blue. Let's make a note that one divided by Tau is equal to 4.29 multiplied by 10 to the power of 11 and its units are collisions per second. And this is the number of collisions an electron makes in one second. So this is an important mental note. So moving right along let's solve for part B. So for part B let us recall and note that the mean free time for copper is about 2.4 multiplied by 10. The power of negative 14 seconds at room temperature. Also recall that resistivity is heavily depends on the charge carriers concentration value rather than the mean free time value. So the concentration of charge carriers in copper is eight 0.5 multiplied by 10 to the power of 28 free electrons per meters cubed. So therefore, high resistivity is linked to a low concentration of available charge carriers. So let's look at our multiple choice sensors to see what answer matches that logic. So we know that our answer for part A is 2.37. So there's only two answers C and D but we know it can't be C because it's not inversely proportional to the mean free time. So that means that the correct answer has to be the letter D. So A is 2.37 multiplied by 10 to the power of negative 12 seconds. And B is geranium has a low concentration of charge carriers. So that is our final answer. Thank you so much for watching. Hopefully, that helped. And I can't wait to see you in the next video. Bye.

Key Concepts

Mean Free Time

Resistivity

Band Gap

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