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Exercise 18.4 the Ideal Gas Law Derived

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Exercise 18.4 the Ideal Gas Law Derived 9/13/2015 Ch 18 HW Ch 18 HW Due: 11:59pm on Monday, September 14, 2015 To understand how points are awarded, read the Grading Policy for this assignment. Exercise 18.4 ∘ A 3.00­L tank contains air at 3.00 atm and 20.0 C. The tank is sealed and cooled until the pressure is 1.00 atm. Part A What is the temperature then in degrees Celsius? Assume that the volume of the tank is constant. ANSWER: ∘ T = ­175 C Correct Part B If the temperature is kept at the value found in part A and the gas is compressed, what is the volume when the pressure again becomes 3.00 atm? ANSWER: V = 1.00 L Correct The Ideal Gas Law Derived The ideal gas law, discovered experimentally, is an equation of state that relates the observable state variables of the gas­­pressure, temperature, and density (or quantity per volume): pV = NkBT (or pV = nRT ), where N is the number of atoms, n is the number of moles, and R and kB are ideal gas constants such that R = NA kB, where NA is Avogadro's number. In this problem, you should use Boltzmann's constant instead of the gas constant R. Remarkably, the pressure does not depend on the mass of the gas particles. Why don't heavier gas particles generate more pressure? This puzzle was explained by making a key assumption about the connection between the microscopic world and the macroscopic temperature T . This assumption is called the Equipartition Theorem. The Equipartition Theorem states that the average energy associated with each degree of freedom in a system at T 1 k T k = 1.38 × 10−23 J/K absolute temperature is 2 B , where B is Boltzmann's constant. A degree of freedom is 1 mv2 m a term that appears quadratically in the energy, for instance 2 x for the kinetic energy of a gas particle of mass Loading [MathJax]/jax/output/HTML­CSS/fonts/TeX/AMS/Regular/Main.js with velocity vx along the x axis. This problem will show how the ideal gas law follows from the Equipartition Theorem. https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=3740013 1/21 9/13/2015 Ch 18 HW To derive the ideal gas law, consider a single gas particle of mass m that is moving with speed vx in a container with length Lx along the x direction. Part A Find the magnitude of the average force ⟨Fx ⟩ in the x direction that the particle exerts on the right­hand wall of the container as it bounces back and forth. Assume that collisions between the wall and particle are elastic and that the position of the container is fixed. Be careful of the sign of your answer. Express the magnitude of the average force in terms of m, vx , and Lx . Hint 1. How to approach the problem ⃗ From the relationship between applied force and the change in momentum per unit time, F = dp/⃗ dt, it follows that the average force in the x direction exerted by the wall on the particle is ⟨Fx ⟩ = Δpx /Δt, where Δpx is the change in the particle's momentum upon collision with the wall and Δt is the time interval between collisions with the wall. You want to find the force exerted by the particle on the wall. This is related to the force of the wall on the particle by Newton's 3rd law. Hint 2. Find the change in momentum Find Δpx , the change in momentum of the gas particle when it collides elastically with the right­hand wall of its container. Express your answer in terms of m and vx . Hint 1. Finding the final momentum The formula for the momentum of a particle p ⃗ of mass m traveling with velocity v ⃗ is p⃗ = mv.⃗ What is the x component of the final momentum of the gas particle (i.e., after the collision)? Express your answer in terms of m and vx . ANSWER: https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=3740013 2/21 9/13/2015 Ch 18 HW pfx = −mvx ANSWER: Δpx = −2mvx Hint 3. Find the time between collisions Use kinematics to find Δt, the time interval between successive collisions with the right­hand wall of the container. ANSWER: 2L Δt = x vx ANSWER: 2 mvx ⟨Fx ⟩ = Lx Correct Part B Imagine that the container from the problem introduction is now filled with N identical gas particles of mass m. The 2 particles each have different x velocities, but their average x velocity squared, denoted ⟨vx ⟩, is consistent with the Equipartition Theorem. Find the pressure p on the right­hand wall of the container. Express the pressure in terms of the absolute temperature T , the volume of the container V (where V = Lx Ly Lz), kB , and any other given quantities. The lengths of the sides of the container should not appear in your answer. Hint 1. Pressure in terms of average force The pressure is defined as the force per unit area exerted on the wall by the gas particles. The area of the right­hand wall is A = Ly Lz. Thus, if the average force exerted on the wall by the particles is ⟨Fx ⟩, then the pressure is given by ⟨Fx ⟩ p = . LyLz Hint 2. Find the pressure in terms of velocity Find the pressure p1 on the right­hand wall due to a single particle whose squared speed in the x direction is 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=3740013 3/21 9/13/2015 1 Ch 18 HW 2 vx . Express your answer in terms of vx , m, Lx , Ly , and Lz. ANSWER: 2 mvx p1 = Lx LyLz Hint 3. Find pressure in terms of temperature 2 To find the pressure from particles with average squared speed vx , you can use the Equipartition Theorem. Find the pressure p1 due to a single particle. Express the pressure due to a single particle in terms of kB , T , Lx , Ly , Lz, and any other given quantities. Hint 1. Relate velocity and temperature 2 Use the Equipartition Theorem to find an expression for m⟨vx ⟩. Express your answer in terms of the gas temperature T , kB , and given quantities. ANSWER: 2 m⟨vx ⟩ = kB T ANSWER: kBT p1 = Lx LyLz ANSWER: p N k T = V B Correct Very good! You have just derived the ideal gas law, generally written pV = NkBT (or pV = nRT ). This applet shows a small number of atoms in an ideal gas. On the right, the path of a specific atom is followed. Look at this for different temperatures to get a feel for how temperature affects the motions of the atoms in an ideal gas. Part C Which of the following statements about your derivation of the ideal gas law are true? https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=3740013 4/21 9/13/2015 Ch 18 HW Check all that apply. ANSWER: 2 2 The Equipartition Theorem implies that ⟨vx ⟩ = ⟨vy ⟩. 2 2 ⟨vx ⟩ = ⟨vy ⟩ owing to inelastic collisions between the gas molecules. With just one particle in the container, the pressure on the wall (at x = Lx) is independent of Ly and Lz. With just one particle in the container, the average force exerted on the particle by the wall (at x = Lx) is independent of Ly and Lz. Correct Part D If you heat a fixed quantity of gas, which of the following statements are true? Check all that apply. ANSWER: The volume will always increase. If the pressure is held constant, the volume will increase. The product of volume and pressure will increase. The density of the gas will increase. The quantity of gas will increase. Correct Up, Up, and Away Hot air balloons float in the air because of the difference in density between cold and hot air. Consider a balloon in which the mass of the pilot basket together with the mass of the balloon fabric and other equipment is mb . The volume of the hot air inside the balloon is V1 and the volume of the basket, fabric, and other equipment is V2. The absolute temperature of the cold air outside the balloon is Tc and its density is ρc . The absolute temperature of the hot air at the bottom of the balloon is Th (where Th > Tc). The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same here. Assume that we can treat air as an ideal gas. Use g for the magnitude of the acceleration due to gravity. Part A What is the density ρh of hot air inside the balloon? Assume that this density is uniform throughout the balloon. Express the density in terms of Th , Tc, and ρc . Hint 1. Find density in terms of temperature and pressure https://session.masteringphysics.comp/Vmy=ct/ansRsigTnmentPrintView?assignmentID=3740013 5/21 9/13/2015 Ch 18 HW Use the ideal gas law, pV = nRT , to find an expression for the density ρ of an ideal gas in terms of its temperature and pressure. Express the density in terms of T , p, R, and m, the mass of one mole of gas. Hint 1. Find density in terms of mass and volume Derive an expression for the density ρ of an gas in terms of the volume occupied by the gas, V , the number of moles of gas particles, n, and the mass per mole, m. ANSWER: ρ mn = V ANSWER: mp ρ = RT Hint 2. How to use your general density equation You now have an expression for the density of a gas in terms of temperature, mass, and pressure.
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