Molecular Motion and Gas Laws What is the connection between the motion of molecules (F = ma and K = mv 2 /2) and the thermodynamics of gases (pv = nrt and U = 3nRT/2)? In this lab, you will discover how the pressure and temperature of air depend on the velocity and kinetic energy of a nitrogen molecule. In previous labs, you studied N = 1 or 2 objects, each of mass m 1 kg, moving at speed v 1 m/s. In today s lab, you will study N 10 23 objects, each of mass m 10 26 kg, moving on average at v 500 m/s! Welcome to the world of atoms and molecules. 1. The Spring of Air In his pioneering memoir on The Spring of Air published in 1660, Robert Boyle introduced the concept of pressure into science. The celebrated ideal gas law, pv = nrt, originated from Boyle s experiments on air pressure. On your table is a cylinder fitted with a piston. Pull the piston all the way to the top of the cylinder. This procedure draws air into the cylinder. The tube connected at the base of the cylinder contains a one-way valve that allows air to flow into the cylinder by not out. Release the piston. The piston should remain at rest at a height somewhere between 95 mm and 100 mm. If the piston falls down, then air is escaping from the cylinder. Consult with your instructor if the cylinder seems to leak. CAUTION: Do not compress the gas too much. High pressure can break parts and shatter plastic. Push down lightly on the piston and feel the springiness of the substance inside the cylinder. Pushing down on the invisible air inside the cylinder feels the same as pushing down on a stiff invisible spring. In this lab, you will measure the spring constant of air. Boyle imagined air to consist of a heap of little bodies, lying one upon another, as may be resembled to a fleece of wool. This pile of bodies compresses when pushed and springs back when released. We now know that the little bodies that make up air and all other gases are molecules in random motion. 2. Molecular Collisions The pressure of a gas is due to the perpetual bombardment of the gas molecules against a surface. The magnitude of the pressure is equal to the average force exerted by the molecules per unit area. Here you will use this molecular-kinetic picture to answer the following basic question: What happens to the pressure of a gas if its volume is reduced to half its original value? The temperature of the gas is kept fixed (at a value equal to room temperature) as the volume is cut in half. Based on your qualitative observation of the springiness of air, it is clear that as the volume of the gas is decreased, the gas pressure increases. To understand the pressure-volume relation in terms of molecular motion, it is sufficient to focus on the motion of one molecule bouncing back and forth at constant speed between the top and bottom walls of the cylinder. By Newton s second law, the average 1
force exerted by the molecule on the top wall (piston) is proportional to the rate at which the molecule hits the wall: force number of collisions per second. The force is also proportional to the change in momentum of the molecule during each collision, but this velocity-dependent quantity depends only on temperature and thus remains constant during the change in volume. Molecular Data: The gas (air) inside your cylinder consists of a multitude of tiny nitrogen balls moving randomly in the x, y, z directions. The mass of one nitrogen molecule is m = 4.65 10 26 kg. The average speed of each molecule at room temperature is v = 515 m/s or 1150 mi/hr. The average velocity component of each molecule is v x = v y = v z = 297 m/s. Note: Air consists of 80% nitrogen (N 2 ) and 20% oxygen (O 2 ). Since the mass of N 2 is close to the mass of O 2, it is okay to assume that air is all N 2. Speed squared is v 2 = v x 2 + v y 2 +v z 2. Since room temperature remains fixed, the average speed of each molecule remains constant. Calculating the Collision Rate. The picture below shows one air molecule (nitrogen ball) bouncing between the ends of the cylinder in your experiment. If the volume is reduced by one half, V V/2, then the rate at which the molecule hits the piston changes. Calculate the collision rate the number of times the molecule hits the piston in one second before and after the reduction in volume. Volume V Volume V/2 297 m/s 100 mm 297 m/s 50 mm Collision Rate = coll/sec. Collision Rate = coll/sec. 2
Molecular Theory Prediction. Based on your collision-rate analysis and the mechanical relations, Pressure Force Collision Rate, state your prediction: If the volume of a gas (at constant temperature) is reduced by a factor of ½, then the pressure of the gas will increase by a factor of. Generalize your collision-rate analysis. For example, what happens to the collision rate and the pressure p if V is reduced from V to V/4? Circle the general p-v relation: p V p V 2 p 1/V p 1/V 2 3. The Force of the Atmosphere When the piston is stationary, the acceleration of the piston is zero and therefore the net force on the piston must be zero. There are two forces acting on the piston: F atm = Force of atmosphere outside the cylinder pushing down on the piston. F gas = Force of gas (air) molecules inside the cylinder pushing up on the piston. We are assuming that the only external force acting on the piston is due to the atmosphere. There is no friction and no push due to your hand or other object. You can neglect the weight of the piston it is negligible compared to the weight of the atmosphere above the piston! Compute the downward force on the piston due to the atmosphere using the following data: Radius of piston = 16.25 mm. Atmospheric pressure: 1 atm = 1.013 10 5 N/m 2. F atm = N. When the piston is at the top of the cylinder (after you pull the piston up to draw in air), what is the upward force of the gas molecules on the piston? F gas = N when L = mm. 3
4. Measuring the Relation Between F gas and L If you push down on the piston with a force probe, the gas will be confined to a smaller space, and therefore the gas molecules will hit the piston at a greater rate: If L decreases, then F gas increases. Note that F atm stays the same. The following force diagrams illustrate this inverse relation between the molecular force F gas and the size of the gas L. F atm F atm F probe F gas L F gas L Here you will use the force probe to study the relation between F gas and L. Pull the piston to the top of the cylinder. Release the piston. The piston should remain at rest at a value of L somewhere between 95 mm and 100 mm. Record this initial value of L in the table below. Set the switch on the force probe to 50 N. Open the file Force Probe. Zero the force probe while holding it in the vertical position. Activate the force probe and then use the probe to push down on the piston through a sequence of heights as follows: (1) Push down quickly until you reach 90 mm. Hold at 90 mm for 2-3 seconds. (2) Push from 90 mm to 85 mm quickly. Hold at 85 mm for 2-3 seconds. (3) Push from 85 mm to 80 mm quickly. Hold at 80 mm for 2-3 seconds. (4) Push from 80 mm to 75 mm quickly. Hold at 75 mm for 2-3 seconds. The whole pushing-down process should take no more than 15 seconds. This short time is necessary to minimize any air leakage from your cylinder. You may have to practice this sequence of pushes-andholds until you get the timing just right. Use Autoscale to see the whole Force-time graph. Your Force-time graph should resemble a staircase of five steps. Each horizontal line segment on your graph, which lasts for 2-3 seconds, corresponds to a different force one for each value of L. Record the Force values in the table below. Repeat this whole procedure pull the piston to the top, push the piston through the sequence of heights, record the forces. Average the two force values. L (mm) 90 85 80 75 F probe (N) 0 F probe (N) 0 Average F probe (N) 0 4
The quantity of interest is the force F gas of the gas molecules inside the cylinder pounding against the piston. Use Newton s second law (net force on piston due to gas + probe + atm must equal zero) to find the value of F gas for each of your four values of L. See the free-body diagrams on the previous page. Show your Newton-law calculations and record your results in the table. L (m) 0.090 0.085 0.080 0.075 F gas (N) 5. Finding the Equation of State Do your experimental values of F gas and L follow a simple pattern? In theory, ideal gases obey the following law of physics: Boyle s Law: The pressure of a gas kept at constant temperature is inversely proportional to the volume of the gas: p 1/V. Hence, the product of pressure and volume is a constant: pv = constant ( for constant T ). So in theory, when you push down on the piston (make L smaller) in your experiment, the pressure of the gas in the cylinder should increase (F gas gets bigger) in such a way that the product F gas L stays the same. Discovering Boyle s Law. Is F gas L equal to a Constant? Compute the product F gas L using your measured values of F gas and L in the previous table: F gas L =,,,,. Are your five values of F gas L equal within experimental error (10%)? If so then you have proven Boyle s Law. If not, then see your instructor. Report the average value of Boyle s constant : F gas L = Joules. This relation between the state variables, F gas and L, is the equation of state of the gas in your cylinder. Express the equation of state in terms of the state variables p and V. Hints: The definition of pressure p is F/A. The volume V of a cylinder is AL. pv = Joules. 5
6. pv Diagram The subject of thermodynamics is full of pv diagrams. Here you will plot F gas vs L, which is equivalent to a p vs V plot. Use the program graphical analysis: plot L on the x axis and F gas on the y axis. Fit your data points with the inverse function y = A/x. Hand in your graph showing the best-fit curve. Answer the following two questions directly on the graph: (1) Compare the value of the curve-fit parameter A with your experimental value of Boyle s constant. (2) Can you conclude that the gas in your cylinder obeys the ideal gas relation p 1/V? 7. How Many Molecules are in the Cylinder? Use your experimental value of pv, together with the ideal gas law pv = nrt, to deduce the number N of air molecules inside your cylinder. Also find the total mass M of air inside the cylinder. Show your calculations. Hints: The gas constant is R = 8.31 J/K mol. Avogadro s number is N A = 6.02 10 23 molecules per mole. Room temperature is 298 K. N = molecules. M = milligrams. 8. What is the Distance Between Air Molecules? Find the average distance d (in nanometers) between the centers of two neighboring molecules in air (at sea level and room temperature). Hints: For T = 298 K and p = 1 atm, the piston in your cylinder is at L 95 mm. For N particles that are uniformly distributed (equally separated by d) in a region of space, the volume of the region is Nd 3. The diameter of an air molecule is 0.30 nm. How many molecular diameters is d? d = nanometers. d = molecular diameters. 6
9. Kinetic Energy of Molecules How do you measure the velocity of an air molecule? How do you keep track of just one molecule? There are 25,000,000,000,000,000,000 molecules occupying each cubic centimeter of air. The motion sensor would have to be an atomic-size (nanometer) device. It turns out that you have already measured the kinetic energy of a nitrogen molecule using the force sensor! Kinetic theory shows that the work constant F gas L, which you measured with the force sensor, is directly related to the average translational kinetic energy ½ mv 2 of a gas molecule: F gas L = (2N/3) ½ mv 2. Calculate the kinetic energy K of the N air molecules in your cylinder two different ways: Microscopic: K = N ½ mv 2 = ½ ( ) 2 = J. Macroscopic: K = 3/2 F gas L = 3/2 = J. 10. Molecular Meaning of Temperature: Using the force relation pv = nrt, the energy relation above can be written in the form ½ mv 2 = 3/2 kt. This important relation says that the absolute temperature of a gas is proportional to the average translational kinetic energy of a molecule in the gas. Calculate the value of the proportionality constant k in the space below. Pay attention to units. k =. Congratulations. You have discovered one of the fundamental constants of nature known as Boltzmann s Constant. The universal constants of nature are G, c, h, k. These four constants characterize the four main fields of physics: Astrophysics (big objects), Electrodynamics (charged objects), Quantum Mechanics (small objects), and Thermodynamics (many objects). 7
11. Molecular Calculation of Force Now that you know the numerical values of N, L, m, and v, you can calculate the pressure of a gas from a purely mechanical analysis of molecules bouncing off a piston. Newton s second law states Average force = Momentum delivered to piston Number of collisions on piston per collision per second Compute the two terms on the right side of Newton s law using the following guide: Momentum delivered to piston per collision = Change in momentum of one molecule during a collision = mv y ( mv y ) = 2 kg m/s = kg m/s. Number of collisions per second = Collision rate for one molecule (for L = 100 mm) Number of molecules = coll/sec = coll/sec. Conclusion. Calculate the average force on the piston. Calculate the pressure due to this force. F = N. p = N/m 2. Your calculated value of pressure (for L = 100 mm) based on molecular motion should agree with the well-known experimental value of atmospheric pressure. Compare these values. 8