ECE3030 Physical Foundations of Computer Engineering, Fall 2015 Homework 1 to Problems 1-4 As a prerequisite to this course, you should understand basic physical quantities such as the universal gas constant (R), Avagadro s number (N A), and Boltzmann s constant (k); in addition, these have been explained both from an intuitive perspective as well as a mathematical perspective. Rather than request repetition of some of the derivations explained in class, this homework will ask for some calculations using the equations as well as some written explanations intutitively explaining what the equations show. Please also utilize the following standard measurement equivalents: 1 atmosphere (atm) = 1.01325 x 10 5 Pascals (P) 0 degrees Celsius ( C) = 273.15 degrees Kelvin (K) 1 unified atomic mass unit (u) = 1 gram per mole (g/mol) 1) (15) Determine the volume of 5 mol of an ideal gas at a temperature of 0 C and a pressure of 1 atm. NOTE1: 0 C and 1 atm are standard temperature and pressure, commonly abbreviated as STP. NOTE2: please give your answer in liters (l ). NOTE3: YOU ARE REQUIRED TO BOX YOUR FINAL ANSWER. According to the ideal gas law: = Equation 1 Where V is volume, is number of moles (amount of the substance), is the ideal/universal gas constant ( = 8.314 1 1 ), is the pressure of the gass. From Equation 1 = 5.000 8.314 1 1 273.2 1.013 10 = 0.1121 10 10 1 = 112.1 1 = 0.1121 = Note 1: = evidences the consistency of the units. Note 2: 1 = 10, 1 = 1 = 10 = 112.1
2) (20) A sample of air with a mass of 100.00 grams (g), collected at sea level, is analyzed and found to consist of the following gases: nitrogen (N2) = 75.52 g oxygen (O 2) = 23.15 g argon (Ar) = 1.28 g carbon dioxide (CO 2) = 0.05 g plus trace amounts of neon, helium, methane and other gases. You look up the elements in the periodic table and find the following: 14.0067 u for nitrogen (N) 15.994 u for oxygen (O) 39.948 u for argon (Ar) 12.011 u for carbon (C) where we assume an average number of neutrons in each atom. The molecular mass is simply the addition of the unified atomic mass unit (u) value for each atom in the molecule, which yield a grams per mole number. Therefore, you calculate the number of moles of each gas type as follows: n(n 2) = 75.52 g / 2(14.0067 g/mol) = 2.6962 mol n(o 2) = 23.15 g / 2(15.994 g/mol) = 0.7234 mol n(ar) = 1.28 g / (39.948 g/mol) = 0.0320 mol n(co2) = 0.05 g / (44.01 g/mol) = 0.0011 mol with the result that the total number of moles in the 100 g sample is 3.4527 mol. Using the ideal gas law and assuming a temperature of 0 C and a pressure of 1 atm, find the volume of the sample. NOTE: YOU ARE REQUIRED TO BOX YOUR FINAL ANSWER. We use Equation 1 and follow a similar approach for = 3.4527 = 3.4527 8.314 1 1 273.2 1.013 10 = 0.07742 10 10 1 = 77.42 1 = 0.07742 = = 77.42
3) (10) In the previous example, the number of moles of carbon dioxide in the sample was found to be 0.0011 mol. Explain how this number was found including how the number of grams per mole was calculated. Use of clear and unambiguous language is required for full credit for this problem; vague and incomplete sentences, including improper grammar, will result in at best some limited partial credit (and possibly zero points especially if the writing is not legible). Taking 15.994 u for oxygen (O), and 12.011 u for carbon (C), we calculate the amount of CO2 in terms of unified atomic mass units as 2(15.994 u) + 12.011 u = 43.999 u. Since 1 u is also equivalent to 1g/mol, then for CO2 43.999 u = 43.999 g/mol. n(co 2) = 0.05 g / (44.01 g/mol) = 0.0011 mol
4) (15) Using (i) the expression = < > for the average (or root-mean-square) velocity of a molecule and (ii) the equation relating the kinetic energy of a molecule to Boltzmann s constant and temperature, give an equation for in terms of Boltzmann s constant (k), temperature (T) and molecular mass (µ). Show how the units of the right hand side work out to be equivalent to the left hand side. For full credit, show all steps clearly and unambiguously. NOTE1: the final correct answer alone will earn zero points; you are required to start with < > on the left hand side of an equation and show the mathematical steps to result in on the left hand side and a function of k, T and µ on the right hand side. You are further required to show how the units of the right hand side reduce to the units of the left hand side (which are m/s where m = meters and s = seconds). NOTE2: the following physical identies may be helpful (i) Joules = Newtons * meters and (ii) Newtons = (kg*m)/s 2. NOTE3: molecular mass (µ) is measured in kg. Method 1 In classical mechanics, the kinetic energy of a particle with mass µ is defined as = 1 2 Equation 2 Due to the statistical nature of each particle in a gas, the average values rather the instantaneous ones are taken in Equation 2. = 1 2 = 1 2 Equation 3 On the other hand, from lecture we related this expression to the Boltzmann constant k and temperature T as 1 2 = = 3 2 Equation 4 Where the factor 3 accounts for the 3 spatial degrees of freedom (x, y and z direction). From Equation 4, = 3 Equation 5 Applying the definition definition, we obtain = = 3 Equation 6
Method 2 Starting from and using the average of the usual definition of kinetic energy (see Equation 3) we obtain = 2 Equation 7 Taking the expresion derived in the lecture (see Equation 4) = 2 = 2 3 2 = 3 Equation 8 Since it is provided that = < >, we obtain the same expression as in Equation 6. Cheking units = 3 Focusing on the units and omitting the numbers in Equation 6 = / = / = /