HW7 Solutions(1)

Chem 113A Homework #7 Solutions Problem 1 The escape velocity from the Earth’s surface is given by vE = (2gR)1/2 where g is gravitational acceleration (9.80 m s–2) and R is the radius of the Earth (6.37 × 106 m). a) At what temperature will νmp for N2 be equal to the escape velocity? b) How does the answer for part (a) change if the gas of interest is He?

Problem 2 Starting with the Maxwell speed distribution, demonstrate that the probability distribution for translational energy for 𝜀�� ≫ 𝑘𝑇 is given by:

𝑓(𝜀𝑇𝑟 )𝑑𝜀𝑇𝑟 = 2𝜋 �

1

𝜋𝑘𝑇

�

3/2

1/2

𝑒−𝜀𝑇𝑟 /𝑘𝑇 𝜀𝑇𝑟 𝑑𝜀𝑇𝑟

Problem 3 Using the distribution of particle translational energy provided in Problem 2, derive expressions for the average and most probable translational energies for a collection of gaseous particles.

Problem 4 Imagine a cubic container with sides 1 cm in length that contains 1 atm of Ar at 298 K. How many gas–wall collisions are there per second?

Problem 5 You are a NASA engineer faced with the task of ensuring that the material on the hull of a spacecraft can withstand puncturing by space debris. The initial cabin air pressure in the craft of 1 atm can drop to 0.7 atm before the safety of the crew is jeopardized. The volume of the cabin is 100 m3, and the temperature in the cabin is 285 K. Assuming it takes the space shuttle about 8 hours from entry into orbit until landing, what is the largest circular aperture created by a hull puncture that can be safely tolerated assuming that the flow of gas out of the spaceship is effusive? Can the escaping gas from the spaceship be considered as an effusive process? (You can assume that the air is adequately represented by N2.)

Problem 6 The diffusion coefficient for CO2 at 273 K and 1 atm is 1.00 × 10–5 m2 s–1. Estimate the collisional cross section of CO2 given this diffusion coefficient.

Problem 7 A thermopane window consists of two sheets of glass separated by a volume filled with air (which we will model as N2 where κ = 0.0240 J K–1 m–1 s–1). For a thermopane window that is 1 m2 in area with a separation between glass sheets of 3 cm, what is the loss of energy when the exterior of the window is at a temperature of 10°C and the interior of the window is at a temperature of 22°C?

Problem 8 Consider the first-order decomposition of cyclobutane at 438°C at constant volume: 𝐶� 𝐻� (𝑔) → 2𝐶� 𝐻� (𝑔)

a) Express the rate of the reaction in terms of the change in total pressure as a function of time. b) The rate constant for the reaction is 2.48 × 10�� 𝑠 ��. What is the half-life? c) After initiation of the reaction, how long will it take for the initial pressure of C4H8 to drop to 90% of its initial value?

Problem 9 Compute νave, νmp, νrms for O2 at 300 and 500 K. How would your answers change for H2

Problem 10 The vapor pressure of various substances can be determined using effusion. In this process, the material of interest is placed in an oven (referred to as a Knudsen cell) and the mass of material lost through effusion is determined. The mass loss (∆𝑚) is given by ∆𝑚 = 𝑍� 𝐴𝑚∆𝑡 where 𝑍� is the collisional flux, A is the area of the aperture through which effusion occurs, m is the mass of one atom, and ∆𝑡 is the time interval over which the mass loss occurs. This technique is quite useful for determining the vapor pressure of nonvolatile materials. A 1.00 g sample of UF6 is placed in a Knudsen cell equipped with a 100 μm-radius hole and heated to 18.2°C where the vapor pressure is 100 Torr. a) The best scale in your lab has an accuracy of ±0.01 g. What is the minimum amount of time must you wait until the mass change of the cell can be determined by your balance?

b) How much UF6 will remain in the Knudsen cell after 5 min of effusion?

Problem 11 Two parallel metal plates separated by 1 cm are held at 300 and 298 K, respectively. The space between the plates is filled with N2 (σ = 0.430 nm2 and CV,m = 5/2 R). Determine the heat flow between the two plates in units of W cm–2.