The semiconductor industry manufactures integrated circuits in large vacuum chambers where the pressure is 1.0×10^−10 mm of Hg.

b. At T=20°C, how many molecules are in a cylindrical chamber 40 cm in diameter and 30 cm tall?

Question

The semiconductor industry manufactures integrated circuits in large vacuum chambers where the pressure is 1.0×10^−10 mm of Hg.

b. At T=20°C, how many molecules are in a cylindrical chamber 40 cm in diameter and 30 cm tall?

Accepted Answer

Hey, everyone. Let's go through this problem. Researchers in plasma physics use specialized plasma chambers operating at low pressures. The temperature and pressure inside a spherical chamber of radius centimeters are 310 Kelvins and 3100. millimeters of mercury respectively. Calculate the number of molecules inside the chamber. We have four multiple choice options to choose from. Option A 1.4 multiplied by 10 to the power of 18 molecules. Option B 1.9 multiplied by 10 to the power of 20 molecules. Option C 1.6 multiplied by 10 to the power of molecules. And option D 5.7 multiplied by 10 to the power of molecules. So because we're trying to find a relationship between the number of molecules and variables like temperature and pressure. This is a good example of a problem where we'll want to use the ideal gas law. More specifically will want to use the version of the ideal gas law stating that the pressure multiplied by the volume of the gas in a chamber is equal to the number of molecules capital N multiplied by the Boltzmann constant K sub B multiplied by the temperature of the gas. Since we're asked for the number of molecules in the chamber, we want to algebraically solve this for capital N dividing both sides of the equation by the Boltzmann constant. And the temperature, we find that the number of molecules is equal to the pressure multiplied by the volume divided by the Boltzmann constant multiplied by the temperature. Well, let's go through the variables in the equation. Bursa, the Boltzmann constant recall that this is equal to 1.38 multiplied by 10, the power of negative 23 Jews per Kelvin. Next, the temperature is given to us as 310 Kelvins. The radius of the spherical chamber chamber is given to us as centimeters which is 250.25 m. Course, the ideal gas law only includes uh the volume, not the radius. But fortunately, since the problem tells us that it is a spherical chamber, we can find the volume by recalling that the volume of a sphere is equal to four thirds pi multiplied by the cube of the radius. So four thirds pi multiplied by 40.25 m cubed. Putting that into a calculator. We find that the volume of the chamber is 0. 45 cubic meters. The problem also gives us the pressure of the air inside the chamber as 0.95 millimeters of mercury. In order for our units to be consistent. We'll have to convert this to past scales by multiplying it by 33. pascals. Since that's how many pascals there are in a one millimeter of mercury put that into our calculator. And that's equal to a pressure of 12.6656 pascals. So now all that's left for us to do is plug all these values in to the ideal gas law equation We solved earlier to solve it for the number of molecules. So first for P, we put in 12. pascals multiplied by the volume, 0.6545 cubic meters and divided by the Boltzmann constant. 1.38 multiplied by 10 of the power of negative 23 jules per Calvin lastly multiplied by the temperature 310 Kelvins. If we put that into our calculator, then we find a number of molecules of about 1.9 multiplied by 10 to the power of molecules. And so that is the answer to this problem. And if we look at our multiple choice options, we can see that option B is exactly that. So option B is then the answer to this problem. And that is it for this problem. I hope this video helped you out. If you'd like more videos like this, please check out some of our other videos and they will hopefully give you more practice with these types of problems, but that's all for now. I hope you all have a lovely day.

The semiconductor industry manufactures integrated circuits in large vacuum chambers where the pressure is 1.0×10^−10 mm of Hg. b. At T=20°C, how many molecules are in a cylindrical chamber 40 cm in diameter and 30 cm tall?

Verified Solution

Key Concepts

Ideal Gas Law

Pressure Conversion

Volume of a Cylinder