A 1.0 m ✕ 1.0 m ✕ 1.0 m cube of nitrogen gas is at 20℃ and 1.0 atm. Estimate the number of molecules in the cube with a speed between 700 m/s and 1000 m/s.

Video transcript

Hey, everyone. Let's go through this practice problem. A spherical container with a diameter of 50 centimeters is filled with helium gas at a temperature of 25 degrees Celsius and a pressure of two atmospheres estimate the number of helium gas molecules in the container that have a speed between 500 m per 2nd and m per second. Assume that the container is perfectly insulated and that the helium gas behaves like an ideal gas. Consider 2.2% of helium gas molecules have a speed between 500 m per 2nd and 1000 m per second. Option A 3.22 multiplied by 10 to the power of 24. Option B 6.15 multiplied by 10 to the power of 24. Option C 7.9 multiplied by 10 to the power of 22. And option D 5.31 multiplied by 10 to the power of 22. This problem isn't too complicated. We can use the variables we've been given along with the ideal gas law to figure out the total number of molecules for the gas. Then once we figure the total number, we can use the percentage we were given or the amount that are within the range that we're looking for to shorten the total number to the number that just fits in with the 2.2%. First, we can find the volume of the gas by using the fact that we're told the gas is in a spherical container. Recalled that the volume of a sphere is equal to four divided by three, multiplied by pi multiplied by the cube of the radius. We're told that the spherical chamber has a diameter of 50 centimeters. So the radius is going to be half of that or 25 centimeters converting this into meters. So 0.25 m and if we put this into a calculator, then we find a volume for the chamber of about 0. 545 cubic meters. So this is the volume of the gas. We also want to consider the pressure. We're told that there is a pressure of two atmospheres. But in order for this pressure to work with the ideal gas law, this seems to be converted into pascals. So we we convert to pascals and multiplying by 1.13, multiplied by 10 of the power power of five pascals per one atmosphere. We put that into a calculator. We find a pressure of about 202,600 pascals. Finally, the temperature we've been given of the gas is 25 degrees Celsius. In order to make the temperature work with the ideal gas law, this needs to be converted into Kelvins and we can convert to Kelvins by adding 273 we put this into a calculator. Then we find a new temperature of about 298. Kelvins recall that the form of the ideal gas law that includes the number of particles is the form that states that the pressure of the gas multiplied by the volume of the gas is equal to the number of particles multiplied by the Boltzmann constant multiplied by the temperature we're trying to solve for the number of particles M. So let's divide both sides of the equation by K T. And we find that the number is equal to the pressure multiplied by the volume divided by K T. Now let's plug into this the variables that we found earlier. So for P that's 202,600 pascals or V, that's 0.6545 cubic meters K is the Boltzmann constant. So that's 1.38 multiplied by 10 to the power of a negative Jews per Kelvin. And the temperature is 298 Kelvins as we discussed earlier. If we put this into a calculator, then we find a number of particles of about 3. multiplied by 10 to the power of 24. So this is the total number of molecules in the gas. But the problem tells us that only 2.2% of the molecules will be within the speed range we're looking for. So let's take this number of molecules we found and multiply it by the percentage 2.22. So point 022, multiplied by 3.22, multiplied by 10 to the power of 24. If we put this into a calculator, then we find a new number of particles of about 7. multiplied by 10 to the power of 22. And so this is our answer to the problem since we've considered the percentage now and everything. If we look at our multiple choice options, we can see if this agrees with option C. So option C is the correct answer to the problem and that's it for this problem. I hope this video helped you out. If you need more practice, please consider checking out some of our other videos which will give you more experience with these types of problems, but that's it for now. And I hope you all have a lovely day. Bye bye.