Homework answers / question archive / Pressure and Ideal Gases Version 6 This worksheet will take you through an explora6on of the proper6es of an ideal gas: temperature, pressure, and microscopic representa6on

Pressure and Ideal Gases Version 6 This worksheet will take you through an explora6on of the proper6es of an ideal gas: temperature, pressure, and microscopic representa6on

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Pressure and Ideal Gases Version 6 This worksheet will take you through an explora6on of the proper6es of an ideal gas: temperature, pressure, and microscopic representa6on. It will build on what you have been learning in class about collisions, forces, and energy, and on the following model of an ideal gas. • • • • Ideal Gas Model: The molecules move in random direc6ons in straight lines un6l they hit the container walls or each other. The collisions with objects are responsible for the pressure of the gas. The gas molecules do not interact with each other except when they collide. They can be modeled as small, hard spheres. The temperature of the gas is propor6onal to the Pressure and Volume of the Gas PV = NkT 1. Pressure and Collisions Playing with beach balls: Imagine that you have one large beach ball and four smaller tennis balls. The large beach ball is ini6ally at rest, and you toss the small tennis balls at it. a. (5 pts) Describe and/or sketch a situa6on where four bouncy balls hit the beach ball, but the beach ball remains sta6onary. What is the direc6on of the net force on the beach ball in this case (to the leL, to the right, or zero)? This describes the situa5on when the pressure is equal on all sides of an object. b. (5 pts) Describe and/or sketch a situa6on where four bouncy balls hit the beach ball and make the beach ball move to the right fairly quickly. What is the direc6on of the net force on the beach ball in this case (to the leL, to the right, or zero)? This describes the situa5on when the pressure is unequal on an object. Key Idea: Ideal gas molecules exert pressure (pressure=force/area) on their containers by colliding with the walls of the container (just as the bouncy balls collided with the beach ball). Increasing the strength of the collisions and/or the frequency of collisions will increase the pressure on the walls. Ideal Gas in a Rigid Container: Now we consider collisions of an ideal gas with a rigid container. For each of the following instances described in the box provided, choose the appropriate answer. For each situa6on, only one variable is changed from the original situa6on. The length of the arrow represents the velocity where longer arrows indicate a larger velocity. Original situa6on, an ideal gas in a container: c. (5pts) Increasing the number of gas molecules will (Increase/Decrease/Not Change) the pressure. Explain your reasoning. d. (5pts) Increasing the volume of the container will (Increase/Decrease/Not Change) the the pressure. Explain your reasoning. e. (10pts) Decreasing the temperature of the gas molecules will (Increase/ Decrease/Not Change) the pressure. Explain your reasoning. f. (10pts) Changing the type of gas from oxygen to helium Decrease/Not Change) the pressure. Explain your reasoning. will (Increase/ Summary and Preview: In this sec6on, you explored the idea of pressure as collisions of gas molecules against container walls. In the next sec6on you will apply these ideas to a situa6on that includes a vacuum. 2. Vacuums and Partial Vacuums a. (5pts) A vacuum is de?ned as a region with no molecules whatsoever. Explain how you can move water up a straw, assuming a vacuum at the top of the straw. True vacuums do not exist. However, par6al vacuums exist, where the number of molecules is far less than typical. For reference, here is a table with values of par6cle densi6es and pressures at various regions inside and above the Earth. Loca5on N u m b e r / c u b i c Pressure cen5meter Gas at standard temperature and pressure 2.5x1019 100,000 Pa Summit of Mount Everest 1017 33,000 Pa Upper atmosphere 1010 0.1 Pa Inside a vacuum chamber 106 1x10-8 Pa Region between Earth and Sun 11 ~0 Pa b. (5pts) Consider a suc6on cup of the type used to hang ornaments from windows. If possible, locate one in your home, then play around with it, and hang some weight from it. Now explain how it works using ideas about ideal gas and pressure that you have just been discussing. Sketches will likely be very useful here. c. (5pts) You are an engineer working for NASA. Your team has been assigned the task of coming up with a way of making space walks safer for astronauts. A fellow engineer proposes giving the astronauts suc6on cups, so they can “s6ck” to the outside of the space sta6on. Will this work? Discuss in your group and explain why it will or will not work. 3. Application of ideal gas ideas a. (10 pts) Watch the video at hgps://www.stevespanglerscience.com/lab/ experiments/incredible-can-crusher/ When the can was ?ipped over in a tub of water, the can collapsed. What was the purpose of pouring some water into the can before hea6ng? Discuss this amongst your group and explain why the can was crushed. b. The New England Patriots were once accused of chea6ng by reducing the pressure of their footballs in an AFC Championship game. The league requires the balls to be between 12.5 and 13.5 PSIg (gauge pressure). At half-6me, on the ?eld the ball pressure was measured to be about 11.5 PSIg. For this problem, you can assume that the balls started in the locker room at around 72° F before the game, and that the half-6me temperature on the ?eld —where the ball pressure was later measured— was 45°F. Ini6ally there were reports that the ball pressure on the ?eld was 10.5 PSIg, but that turned out to be inaccurate. Useful informa5on : We can use the ideal gas law PV=NkT to inves6gate this: In the locker room Pi V = Nk Ti and on the ?eld Pf V = Nk Tf So we can conclude that Pf = (Tf/Ti ) Pi Note that we must use absolute pressure and absolute Temperature here. Gauge pressure is the pressure di?erence from 1 atm. Absolute pressure is the pressure measured with respect to perfect vacuum. PSI stands for “Pounds per Square Inch” and is a common unit of pressure. To discriminate between Gauge and Absolute pressure the leger “g” or “a” is some6mes added to the end of the pressure unit. For example, “PSIg” or “PSIa”. So 1 Atm = 101.3 kPa = 14.7 PSIa = 0 PSIg In other words, to get from PSIg to PSIa, you need to add the o?set of 14.7 Example :convert 100 PSIg to PSIa: P = 100 +14.7 = 114.7 PSIa. Example : convert 33 PSIa to PSIg: P= 33 - 14.7 = 18.3 PSIg Please keep the di?erence between gauge and absolute pressure in mind when solving the following problem. c. (10 pts) In response to the allega6ons of chea6ng, based on the ini6al rumors of 10.5 PSIg, it was incorrectly concluded that “the pressure dropped from 12.5 PSIg to around 10.5 PSIg, which is a rela6ve drop of 16%. Explain what is wrong with this statement. d. (20 pts) Calculate what the Gauge pressure in PSIg of a football would be when it has cooled down to 45° F, if it was at 12.5 PSIg when at 72° F. You will need to use absolute pressure and temperature in your calcula5ons, but in the end convert back to gauge pressure! e. (5 pts) The Patriots’ footballs were measured to be at 11.5 PSIg (average) at half6me just aLer being outside for hours on a rainy, 45° F day. Based on your calcula6ons above, can you conclude that they were illegally de?ated? Discuss. Name(s):______________________________________________________________________ Section:_______________________________________________________________________ TA: __________________________________________________________________________ Applications of Sine and Cosine in Physics Based on an original exercise by Gillian Galle and Dawn Meredith July 2012 Modified 2021 by kjs This lab takes trigonometric concepts and put them in a physics context. Physical context Picture a coffee cup sitting on the edge of 15 cm radius turntable in a microwave, and the microwave is running, so the coffee cup is moving in a circle at a constant rate, going around once in 30 seconds. If you get down to eye level with the microwave, you will see just the xcomponent of its motion, and (from our knowledge of the unit circle) x=cos(θ). 1. (10 points) To be sure this makes sense, draw a picture of the turntable from the top and draw x-y axes. Pick one location on the circle and indicate both θ and x for that location of the cup. Need for a more refined model The mathematical model x=cos(θ) is not quite correct or complete. First, we want to include in this description how θ changes in time, and also, the turntable is not a unit circle, but a circle of radius 15 cm. In this activity we will learn how to put these refinements in the mathematical model. How θ changes in time. Recall that for an object moving at a constant rate in a straight line (e.g. a car moving down a straight road) x=x0+vt, where x0 is the location at t=0. Analogously, when the cup moves around a circle at a constant rate, θ = θ0 + ω t. (We will take θ0=0 from now on). The notation ω stands for angular velocity. 2. (5 points) Recalling linear motion for a moment, if an object moves 30m in 5 seconds, what is v? v= 3. (15 points) Using the definition that ω = (change in angle)/(change in time), what is ω for the cup on the turntable? Be sure to use radians (not degrees) for the angle. (ω is conventionally measured in radians/s .) ω= Using ω, what angle has the cup traveled to at 15 seconds? Show your work for full credit and be sure to keep units. As a check, what fraction of a cycle is 15 seconds? Does that agree with your angle? θ (15 s) = Using ω, what angle has the cup traveled in 600 seconds? Show your work for full credit and be sure to keep units. As a check, how many cycles is 600 seconds? Does that agree with your angle? θ (600 s) = Verify that the last two equations have correct units for an angle. If not, look for the mistake, or check with the instructor. 4. (20 points) We end this task by finding a symbolic way of writing the angular velocity that corresponds to an angular displacement of 2 pi during an unspecified period T seconds. What is the ω in this case? ω= Using ω, what angle has the cup traveled in (T/4) seconds? θ= Now draw the motion along the turntable for the last example. obtained the correct angle above. Use this to check that you Lastly, using angular velocity, what angle corresponds to t seconds, where t is an unspecified time? Your answer will have both T and t – these are not the same. “T” stands for a fixed but unspecified time to go around the circle once ; “t” stands for the variable time; think of it as all the various readings of your watch as the cup goes around the turntable. θ (t)= (This is boxed because this general formula will be useful many times as you learn about oscillations and waves.) 5. (15 points) Now that we’ve seen how θ depends on time for uniform circular motion, let’s see how that works in conjunction with the sine and cosine function. In the table below, “s” stands for seconds. What is the period of the motion, and how can you tell based on the equations for x(t) in the table? By filling out the “fraction of a cycle” row first, you should be able to fill out the other two rows without a calculator. t= Fraction of a cycle ? 2π t ? x(t) = sin ? ? ? (5s) ? ? 2π t ? x(t) = cos ? ? ? (5s) ? 0s 1.25 s 2.5s 3.75s 5s 6.25s 7.5s 8.75s 10s What about the 15 cm radius? 6. (20 points) Lastly, we need to take into account that the cup travels on a 15 cm radius turntable, and not the unit circle. Which one of the following mathematical models describes that motion? Choose and then fill out only the correct table. Cross out the other three tables, but explain in words why the model fails or works in each case. For this problem, consider the case where the coffee completes a revolution in 5 seconds. t= 0s 1.25 s 2.5s 3.75s 5s 0s 1.25 s 2.5s 3.75s 5s Fraction of a cycle t= Fraction of a cycle t= 0s 1.25 s 2.5s 3.75s 5s t= 1.25 s 2.5s 3.75s 5s Fraction of a cycle t= Fraction of a cycle 7. (15 points) As a check on your understanding, sketch the graphs of the following function. Include two full periods and label the tick marks along both the axes. (In the equations below “m” is for meters and “s” is for seconds.) Note that these are NOT the position and velocity for the same object. Use what you learned in today’s activity to make graphing easy; you should not need a calculator. (a) position : Graph x vs t here. Period: ______________________________ Maximum Value:_______________________ Minimum Value:_______________________ x-intercept(s):__________________________