CONSERVING EQUATIONS

Next: Homework Assignment for Week 4 Previous: Homework Assignment for Week 2

Problems from the subsection titled ``More on the Derivative and Differential Equations''

3.1:
- a.) Use Maple to plot x squared and 2x on the same axes for x=-3..3. Verify that the derivative has the properties you would have expected it to have (i.e., it is positive when f is increasing, etc..);
- b.) In Maple, define Remember: In Maple, functions are defined using f:=x-> notation, and not f(x):= notation!). Define a gav function for this f, using gav:=(x,h) -> (f(x+h)-f(x))/h and display the expression expand(gav(x(h)). Notice that there is no h in the denominator of this expression. Then substitute h=0 into this last expression (Maple has a subs(h=0,") command) and compare this with diff(f(x),x).
- c.) Using the function from part (b), plot both f(x) and f'(x) on the same axes over the interval [-1,2]. Which is the graph of the function and which is the derivative? How do you know?
3.2:
Use Maple to verify the sum rule for the functions and
3.3:
Use Maple to verify the product rule for the functions and
3.4:
Verify the stretch rule for the functions .
3.5:
- a.) State versions of the sum, product, constant multiple, and stretch rules for second derivatives.
- b.) Use your stretch rule to show that the function e^{rx} (for some constant r) is a solution of the differential equation
```
			ay'' + by' + cy = 0
    
```
  if and only if ar^2 + br + c = 0.
- c.) Find solutions of the differential equations y'' + 15y' + 8y = 0 and y'' - 15y' + 8y = 0 by hand. Then use Maple's dsolve command to do the same thing --- part of this problem is to figure out how to use dsolve -- remember to look at and try out the examples in the ``Help'' menus!
Problems from the subsection titled ``Force Revisited''
3.6:
Imagine a landing craft approaching the surface of Callisto, one of Jupiter's moons. If the engine provides an upward force (thrust) of 3260 N, the craft descends at constant speed; if the the engine provides only 2200 N, the craft accelerates downward at 0.39 m/s^2.
- a.) What is the weight of the landing craft in the vicinity of Callisto's surface?
- b.) What is the mass of the craft?
- c.) What is the free-fall acceleration near the surface of Callisto?
3.7:
A 10 kg monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a 15 kg package on the ground.
- a.) What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground?
  If, after the package is lifted, the monkey stops its climb and holds onto the rope, what are
- b.) its acceleration?
- c.) the tension in the rope?
3.8:
An elevator operates by having a passenger cage (labeled A in the figure below) attached to a pulley, a driving mechanism (labeled C), another pulley, and a counterweight (labeled B). In operation, mechanism C grabs the cable, either forcing it along or retarding its motion. This process means that the tension T_A in the cable attached to the passenger cage does not have to be the same as the tension T_B attached to the counterweight. Suppose the upward acceleration of A and the downward acceleration of B have the magnitude a = 2.0 m/s^2. The mass of the elevator cage is 1100 kg and the mass of the counterweight is 1000 kg. Neglecting the pulleys and the mass of the cable, find
- a.) the tension T_A (hint: the answer is 1.3 x 10^4 N)
- b.)the tension T_B (hint: the answer is 7.8 x 10^3 N);
- c.) the force on the cable produced by C. (hint: the answer is 5.2 x 10^3 N)
3.9:
- a.) Find the acceleration of masses m_1 and m_2 for the figure shown below. The pulleys are massless and all surfaces are frictionless.
- b.) What do these results predict in the limits m_2 >> m_1 and m_1 >> m_2?
  
  Hint: the answers are
- 3.10:
  The most general case of elastic collisions in one dimension in which all velocities lie along a single axis can be solved generally and the general solution adapted for special cases. Consider a one-dimensional elastic collision on a frictionless surface for two objects, say billiard balls 1 and 2, with masses m_1 and m_2. Object 1 has initial velocity u_1 and object 2 has initial velocity u_2. After the collision, objects 1 and 2 have final velocities v_1 and v_2, respectively. Sketch two pictures which indicate two possibilities for the directions of the initial velocities of 1 and 2 such that a collision will eventually occur.
  - a.) Write down the equations for momentum and kinetic energy conservation for this situation, then use Maple to solve them to find expressions for the final velocities, v_1 and v_2.
  - b.) Assume that the initial velocities are known for both objects. Use your Maple solutions for the final velocities to find v_1 and v_2 in terms of the initial velocities in the case in which the masses of the objects are identical. What is your result? Write down in words what your expressions for v_1 and v_2 actually tell you happens in elastic collisions of identical mass objects such as billiard balls.
  - c.) Assume now that m_1 and m_2 are different and that object 2 is initially stationary while object 1 has a known initial velocity directed toward object 2. Use your Maple solutions for v_1 and v_2 from part a to find expressions for v_1 and v_2 in terms of u_1 for this case. Once you have these expressions, you can make plots which show how the two velocities depend on the relative values of m_1 and m_2 by assigning m_2 in Maple to be a function of m_1, e.g. m2 := x*m1 and assigning u_1 to have a value 1 so that your plots of v_1 and v_2 are in units of u_1. Do two plots, one for x=1..10 (i.e. m_2 is 1 to 10 times as large as m_1) and one for x=0.1..1 (m_2 is 1/10 of m_1 to identical to m_1). Summarize your conclusions about each plot by describing in words what happens to objects 1 and 2 after the collision, e.g. what happens when the objects are the same mass? when m_2 is much smaller than m_1? when m_2 is much larger than m_1?
- 3.11:
  - a.) You can use your results from the previous exercise in which you wrote down equations for the final velocities of dissimilar mass objects after a collision to estimate the fraction of kinetic energy lost by a neutron in a head-on elastic collision with a nucleus of mass m_2. Assume that m_2 is initially at rest. (Hint: Begin by writing down what the fractional reduction in kinetic energy is given initial velocity u_1 and final velocity v_1 for the neutron, then find the ratio u_1/v_1 using your results from part c. of the previous exercise.)
  - b.) The reason for using a material with light nuclei becomes obvious when you take your result from part a and substitute in real numbers. Evaluate the fraction you derived in part a for lead, carbon, and hydrogen nuclei. The ratios of nuclear mass to neutron mass (i.e. m_2/m_1) for these nuclei are 206 for lead, 12 for carbon, and about 1 for hydrogen.
- 3.12:
  The Voyager 2 spacecraft used the slingshot effect around Jupiter to propel it to its next target, Saturn. To get some idea of the power of this effect, assume that the probe has mass m_1 and initial speed u_1 as it approaches Jupiter (mass m_2 and speed u_2 relative to the sun as shown in the figure below). The spacecraft rounds the planet and departs in the opposite direction. What is its speed, relative to the sun after this encounter? You can assume that u_1 = 12 km/s and u_2 = 13 km/s (the orbital speed of Jupiter). You must also assume that the mass of Jupiter is much greater than the mass of the spacecraft.
- 3.13:
  A Saturn V rocket like the one that took men to the moon has a mass of 2.5 x 10^6 kg on liftoff. It goes straight up vertically and burns fuel at a uniform rate of 1.6 x 10^4 kg/s for a duration of 2 minutes. The exhaust speed of gas from the Saturn V is 3.0 km/s.
  - a.) What is the speed of the rocket immediately after the combustion ceases? You should include the effect of gravity near the surface of the earth, but you can neglect air resistance.
  - b.) Plot the burnout velocity as a function of time over the range of 0 to 120 seconds to see the increase in speed of the rocket with time. Does this graph seem to have the qualitative aspect you expect compared to the graph of speed vs. mass generated earlier in the Maple example in your lecture notes? If not, why do your graph and the one generated in the Maple example appear different?
  - c.) The speed necessary to achieve a stable Earth orbit is about 7.9 km/s. Does your rocket make it to Earth orbit or not? To make it easier to boost payloads to orbits, most rockets are multi-stage. Each stage propels only the weight of fuel and rocket body for things above it. After a stage is burnt out, it is dropped and the next stage ignites. The velocity increase of the new stage is added to that of the old, thereby making a more efficient use of rocket weight and allowing very high speeds. Write down an equation which describes the velocity of a two-stage rocket as a function of time. Assume that the mass of the rocket at liftoff is M_0, the mass of fuel in the first stage is M_1, M_1' is the mass of the first stage that is discarded, and M_2 is the mass of the second-stage fuel. Assume that gravity is constant.
  Problem from the subsection titled ``Second-Order Differential Equations''
- 3.14:
  - a.) Maple's dsolve can handle initial-value problems, too. For the example we just did, enter:
```
	dsolve(diff(x(t),t$2)+9*x(t)=0,x(0)=1,D(x)(0)=6},x(t));
  
```
    Try it. Plot the solution. To what extent is it what you expected physically. How does it depart from reality?
  - b.) To put in a friction force that is proportional to, but in the opposite direction from the velocity, how should the equation change? Make up and solve your own examples for various values of the friction proportionality constant. Plot the results. Try enough different values of the constant so that the behavior of the solutions are qualitatively different. Turn in your results (problems, initial conditions, and formulas and plots of the solutions) for three values of the friction constant that yield qualitatively different solutions. (More on this next week!!)
  Problem from the subsection titled ``What's Your Reaction?''
- 3.15:
  A certain overall reaction in aqueous solution, has the form:
  
  The initial rate of this reaction was measured at 25° C, as a function of initial concentrations (in M) of A, B, and C. The data are as follows:
  
  Trial [A]_0 [B]_0 [C]_0 Initial Rate (in M/s)
  
  #1 0.02 0.02 0.02 1.414
  
  #2 0.06 0.02 0.02 12.726
  
  #3 0.06 0.06 0.02 4.242
  
  #4 0.03 0.06 0.03 1.299
  - a.) Deduce the experimental rate law. Find the order with respect to A, with respect to B, with respect to C, and the overall order.
  - b.) Calculate the experimental rate constant, k_{exp}, in the appropriate units.
  - c.) If equal volumes of 0.100 M aqueous solutions of each reactant are mixed at t = 0, find the time that it takes B to decrease to a value of 2.57 x 10¯ ³ M. What percent of B has been consumed at this point?
- 3.16:
  Using Maple, do the following for a stoichiometric reaction of the form A --» Products; Assuming that the reaction follows first-order kinetics, i.e., Rate = - d[A]/dt = k[A]:
  - a.) Solve the differential rate law for [A] as a function of time (t) in terms of the initial amount of (A(0)) and the rate constant (k).
  - b.) Using the "seq" command Plot [A] versus t for a range of A(0) values (say, from 0.5 M to 2.0 M) - allowing k to be fixed at 1.0 s¯ ¹ . Then for A(0) = 2.0 M, plot the [A] versus t for a range of k values (say, k = 1.0 s¯ ¹ , 2.0 s¯ ¹ , 3.0 s¯ ¹ , 4.0 s¯ ¹ , 8.0 s¯ ¹ or choose your own). Comment on what you see in each set of plots.
- 3.17:
  Using Maple, do the following for a stoichiometric reaction of the form A --» Products; Assuming that the reaction follows second-order kinetics, i.e., Rate = - d[A]/dt = k[A]² :
  - a.) Solve the differential rate law for [A] as a function of time (t) in terms of the initial amount of (A(0)) and the rate constant (k).
  - b.) Using the "seq" command Plot [A] versus t for a range of A(0) values (say, from 0.5 M to 2.0 M) - allowing k to be fixed at 1.0 s¯ ¹ . Then for A(0) = 2.0 M, plot the [A] versus t for a range of k values (say, k = 1.0 s¯ ¹ , 2.0 s¯ ¹ , 3.0 s¯ ¹ , 4.0 s¯ ¹ , 8.0 s¯ ¹ or choose your own). Comment on what you see in each set of plots.
  - c. Determine the formula (in terms of A0 and k) for the half-life (t_1/2) for a reaction that follows second order kinetics - using the A(t) expression determined in part (a). You may use Maple - as usual - to help you. Explain carefully what should happen to the half-lives - and why - as an initial amount of A (A0) is decreased to 1/2, 1/4, 1/8,... of its initial amount.
- 3.18:
  Using Maple, do the following for a stoichiometric reaction of the form A --» Products; Assuming that the reaction follows zero-order kinetics, i.e., Rate = - d[A]/dt = k[A]° = k:
  - a.) Solve the differential rate law for [A] as a function of time (t) in terms of the initial amount of (A(0)) and the rate constant (k).
  - b.) Using the "seq" command Plot [A] versus t for a range of A(0) values (say, from 0.5 M to 2.0 M) - allowing k to be fixed at 1.0 s¯ ¹ . Then for A(0) = 2.0 M, plot the [A] versus t for a range of k values (say, k = 1.0, 2.0, 3.0, 4.0, 5.0 or choose your own). Comment on what you see in each set of plots.
  - c.) What kind of chemical process, i.e. under what circumstances might a chemical reaction follow zero-order kinetics? Explain your answer.
  - d. Determine the formula (in terms of A0 and k) for the half-life (t_1/2) for a reaction that follows zero order kinetics - using the A(t) expression determined in part (a). You may use Maple - as usual - to help you. Explain carefully what should happen to the half-lives - and why - as an initial amount of A (A0) is decreased to 1/2, 1/4, 1/8,... of its initial amount.
- 3.19:
  Consider a reaction of the form: X(g) --» Products, where it is believed that the rate law has the form:
  -d[X]/dt = k[X]^n
  where n is unknown. The following data is collected. Note that time progresses as one reads down the columns. So, the initial concentration of X is 0.50 M.
```
     [X] (in M)      Initial Rate of depletion of [X] = 
     0.50              300.0
     0.30               64.8
     0.25               37.5
     0.20               19.2
     0.15                8.1
```
  Use Maple to answer the following parts:
  1. Use the van't Hoff method to determine the value of the order "n" and the value of the rate constant k - in units of M and seconds. Be sure to plot your data along with the best-fit line on the same graph!!
  2. Knowing your value of "n", k, and [X]_0; plot initial rate versus concentration and see if you get the "expected" looking curve according to (a). Explain.
  3. Solve the differential equation that results from the determined rate law - based on the initial conditions. Does your result make sense? Explain.
- 3.20:
  Referring to Example #2 in the section on Mechanism of a Chemical Reaction, do the following.
  1. By suitable use of the steady-state approximation (SSA), prove that the steady-state solution of Y and of Z are as listed. In other words, derive the expressions for [Y(t)]_{ss} and [Z(t)]_{ss}.
  2. Plot the steady-state solutions of [X(t)], [Y(t)], and [Z(t)] on the same graph. Choose the same initial conditions as given in Example #2 and choose values of k_1 and k_2 such that k_2 » k_1 (say 100 : 1). Comment on what you see - as compared to the plot of the exact solutions and the carefully explain in terms of the SSA.
  3. Consider the exact solutions of [X(t)], [Y(t)], and [Z(t)] once more. Determine how the exact solutions will change (using Maple of course) in the initial condistions are changed.
    - Perform Maple plots of [X(t)], [Y(t)], and [Z(t)]Z (on the same graph) for
    - Then, perform Maple plots of [X(t)], [Y(t)], and [Z(t)]Z (on the same graph) for
- 3.21:
  Consider the reaction mechanism of three (3) consecutive unimolecular gas phase reactions as shown:
```
                 
```
  1. Referring to the Maple writeup for solving and plotting systems of differential equations as needed, find the exact solution of [X(t)], [Y(t)], [Z(t)] and [W(t)]. Assume that you begin only with X(g) and nothing else. Hint: An important equation in your system is the "conservation of mass equation", i.e. [X(t)] + [Y(t)] + [Z(t)] +[W(t)] = [X]_0.
  2. Plot [X(t)], [Y(t)], [Z(t)] and [W(t)] on the same graph, assuming that [X]_0 = 2.0 M and pick various values of k_1, k_2, and k_3. Do at least three (3) different plots - varying the k's so that the various features of the solutions in (1) are portrayed.
- 3.22:
  Consider the unimolecular reversible reaction - not necessarily at equilibrium:
```
                  
```
  1. Referring to the Maple handout for solving and plotting systems of differential equations as needed, find the exact solution of [X(t)], [Y(t)] of the following situations. In each case plot [X(t)] and [Y(t)] on the same graph. Assume k_1 = 1.0 s ¹ and k_2 = 0.5 s ¹ throughout. Assume the following situations:
```
                 
```
  2. Extra Credit: How does equilibrium enter into the picture?
- 3.23:
  The free radical reaction of methane (CH_4) to produce ethane (C_2H_6) is believed to occur via the three reversible steps:
```
                 
```
  where k_1 = 14.0 s¯ ¹ , k_2 = 1.2 x 10¹ ° M¯ ¹ s¯ ¹ , k_3 = 1.5 x 10^9 M¯ ¹ s¯ ¹ , k_4 = 2.9 x 10^7 M¯ ¹ s¯ ¹ , k_5 = 2.0 x 10¹ ° M¯ ¹ s¯ ¹ , and k_6 = 4.5 x 10^4 s¯ ¹ . Also, only methane is initially present, at an initial concentration of 1.0 x 10¯ ³ M.
  1. Write down the overall stoichiometric reaction and identify the overall reactants(s), overall product(s), and any intermediates
  2. Write the kinetic equations (rate laws) for this simplified mechanism for the time derivatives of all intermediates, reactant(s), and product(s).
  3. Assuming the validity of the steady-state (SSA) approximation on all free radical intermediates, determine the steady-state expressions for
    and for , i.e. and .
  4. Derive the differential rate law for the formation of as a function of and how ever many elementary rate constants are needed. Make use of your results from (2) and (3) as needed.
  5. Plot the time dependence of radicals and of over a 30 microsecond range .
  6. If k_2 is increased by a factor of 5, what is the percentage increase in the ?
Assignments for Week 3

Due on Tuesday, 8/15: Prob. 2.13
Due on Wednesday, 8/16: Probs. 2.22, 2.23, 3.6, 3.7, 3.8, and 3.9. Chemistry Pre-lab. REMEMBER TO MEET WEDNESDAY AT 1:30 PM in DRL A8!

Next: Homework Assignment for Week 4 Previous: Homework Assignment for Week 2

larryg@upenn5.hep.upenn.edu
Tues Jul 26 13:50:36 EST 1994
hmwrk_week_3.html

Trial	[A]_0	[B]_0	[C]_0	Initial Rate (in M/s)
#1	0.02	0.02	0.02	1.414
#2	0.06	0.02	0.02	12.726
#3	0.06	0.06	0.02	4.242
#4	0.03	0.06	0.03	1.299

CONSERVING EQUATIONS

Problems from the subsection titled ``More on the Derivative and Differential Equations''

Problems from the subsection titled ``Force Revisited''

Problem from the subsection titled ``Second-Order Differential Equations''

Problem from the subsection titled ``What's Your Reaction?''

Assignments for Week 3