You are watching a science fiction movie in which the hero shrinks down to the size of an atom and fights villains while jumping from air molecule to air molecule. In one scene, the hero's molecule is about to crash head-on into the molecule on which a villain is riding. The villain's molecule is initially 50 molecular radii away and, in the movie, it takes 3.5 s for the molecules to collide. Estimate the air temperature required for this to be possible. Assume the molecules are nitrogen molecules, each traveling at the rms speed. Is this a plausible temperature for air?

Question

Accepted Answer

Hey everyone. So this problem is dealing with the root mean squared speed. So let's see what it's asking us. We have a high spatial resolution microscope that is developed to study the motion of gas molecules. And we're looking at oxygen molecules at standard pressure inside a small container. We detect on the microscope, two oxygen molecules separated by a distance of 22 molecular radii which then co collide head on. After 2.3 seconds, we are told to assume that the oxygen molecules have the same routine square speed and that there are no other gasses present in the container. And they tell us that that molecular radii I radius of oxygen is 1.52 angstroms and its molecular weight is 32 g per mole, determine the temperature of the oxygen gas. In Kelvin, our multiple choice answers are a 6.8 times 10 to the negative 22 Kelvin B 3.2 times 10 to the negative five Kelvin C 2.5 Kelvin or D 95 Kelvin. So we can, the first step in solving this problem is recalling that the root means squared speed or V sub R MS is equal to three multiplied by the Boltzmann constant denoted by K. So B multiplied by T our temperature divided by our maps. And so to find that route means squared speed, we are told the distance that the um molecules travel and the time in which they do that. And so that can be found with a simple speed equals distance divided by time equation. And so that will be the first step. So our distance if both molecules traveled 22 molecular radii that, that they were separated originally, that means each traveled 11 molecular radii to meet in the middle at that collision point. And so that's 11 multiplied by 1.52 an angstrom we can recall is a is 10, the negative 10 m. And the time in which they traveled that distance was 2.3 seconds. And so we plug that into our calculator and we get 7. times 10 to the negative 10 m per second. Now, looking at the problem, we're asked to determine the temperature and so we can rewrite this equation, our root mean squared speed equation to isolate temperature as a variable on the left-hand side of the equation. And that becomes temperature is equal to the root mean squared speed squared multiplied by mass all divided by three, multiplied by the bolt constant. And so when we go to plug in those values, we use the root mean square speed that we just saw for 7.27 times 10 to the negative 10 m per second. The mass is going to be 32 U or atomic mass units because one molecule of oxygen um 02 is multiplied by our conversion factor between AM U and kilograms which is 1.66 times 10 to the negative kg per U. All of that will be divided by three multiplied by our bolt constant, which we can recall is 1.38 times 10 to the negative 23. And that is in units of jewels per Kelvin. So when we plug all of that into our calculator, we get a temperature of 6.8 times 10 to the negative 22 Kelvin. And so this is an unrealistic answer for oxygen because oxygen is a liquid at 90 degrees Kelvin. And so that might be another question. Is this physically possible? And so uh if we needed to draw that conclusion, you know, that answer would be no. All right. So that answer choice aligns with answer choice A and that's all we have for this one. We'll see you in the next video.

Verified Solution

Key Concepts

Root Mean Square Speed (rms speed)

Kinetic Theory of Gases

Collision Frequency