The solar corona is a very hot atmosphere surrounding the visible surface of the sun. X-ray emissions from the corona show that its temperature is about 2×10^6 K. The gas pressure in the corona is about 0.03 Pa. Estimate the number density of particles in the solar corona.

Video transcript

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. So charcoal fire can attain temperatures as high as 1200 degrees Celsius. The temperature of the gas escaping from the fire can be assumed to be at 1200 degrees Celsius and a pressure of 1.13 multiplied by 10 to the fifth power past scales calculate the approximate number density of the gas particles in the gas escaping the fire. OK. So we're given some multiple choice answers here and they're all in the same units of we'll highlight it particles per meters cubed. So let's read them off and see what our possible answer might be. A is 8.27, B is 6.11 multiplied by 10 to the 24th power C is 4.98 multiplied by 10 to the 24th power and D is 10.2. So first off, let's assume that the gas escaping the fire is an ideal gas. So then we can recall and use the ideal gas law, which as we know, the ideal gas law is PV equals N RT which with, in this case, it's lower case N. But we can't use this equation means we're trying to find the number density of gas particles. Instead, we have to use the following equation instead. So PV equals capital N multiplied by lowercase K subscript capital B multiplied by T. In this case, it's pressure multiplied by volume is equal to the number of particles multiplied by Boltzmann's constant multiplied by temperature. Also, we need to quickly recall that the number density, which I'm gonna denote it as capital N capital D. The number density equation is as follows N D, the number density is equal to the number of particles divided by the volume. Awesome. So our end goal is to find the number density of particles. So we must rewrite. We're gonna call this equation one, we need to rewrite equation one. We need to rewrite the ideal gas law to solve for the number density. So when we rearrange it to solve for the number density, we should get the following that N D. The number density is equal to the pressure divided by Boltzmann's constant multiplied by temperature. So let's make a quick note that the temperature as given to us in the prom was 1200 degrees Celsius. And that the pressure was 1.13 multiplied by 10 multiplied by 10 to the fifth power past skills. And that Boltzmann's constant, the numerical value for that is 1.381 multiplied by 10 to the minus 23rd power and its units are jewels per Kelvin. Awesome. OK. But we have a little bit of a problem. So we're given the temperature as 1200 degrees Celsius. But it needs to be in Kelvin in order for us to use our equation properly. So let's do a quick unit conversion. So we need to recall how to convert degrees Celsius to degrees Kelvin. And to do that, we take our degrees Celsius, which in this case, it was 1200 degrees Celsius. And all we have to do is add 273. When we do that, our temperature in Kelvin should be Kelvin. So now that we have our temperature in Kelvin, we can finally calculate the number density. So the number density when we plug in all of our numerical values, our pressure was 1.13 multiplied by to the fifth power past scales. And we know that Boltzmann's constant was 1.381 multiplied by 10 to the minus or excuse me, 10 to the negative 23rd power jewels per Kelvin. And then our temperature was 1473. So it's Boltzmann's constant multiplied by the temperature. So when you plug that in all of this into a calculator, you should get 4.98 multiplied by 10 to the 24th power. And the units are particles meters. Cute. And let's make a quick note here in case you're wondering how in the world did past scales turn to this. So as a quick little side note, so pascals are equal to jewels per meters cubed. Awesome. So that's how when we cancel out and everything here, our units means the Kelvins co cancel out and we're left with pascals which is equal to a duel per meters cube. So that's how we get our final unit value there. So, so that would mean our final answer is C 4.98 multiplied by 10 to the 24th particles per meter cube. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video.