Introduction to Chemical Engineering Processes/Mathematical Methods Practice Problems

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Example
Problem:

1. In enzyme kinetics, one common form of a rate law is Michaelis-Menten kinetics, which is of the form:

 -r_S = \frac{V_{max}*[S]}{K_m + [S]}

where  V_{max} and  K_m are constants.

a. Write this equation in a linearized form. What should you plot to get a line? What will the slope be? How about the y-intercept?

b. Given the following data and the linearized form of the equation, predict the values of  V_{max} and  K_m

[S], M   rS, M/s
0.02     0.0006
0.05     0.0010
0.08     0.0014
0.20     0.0026
0.30     0.0028
0.50     0.0030
0.80     0.0036
1.40     0.0037
2.00     0.0038

Also, calculate the R value and comment on how good the fit is.

c. Plot the rate expression in its nonlinear form with the parameters from part b. What might  V_max represent?

d. Find the value of -rS when [S] is 1.0 M in three ways:

  1. Plug 1.0 into your expression for -rS with the best-fit parameters.
  2. Perform a linear interpolation between the appropriate points nearby.
  3. Perform a linear extrapolation from the line between points (0.5, 0.0030) and (0.8, 0.0036).

Which is probably the most accurate? Why?

Example
Problem:

2. Find the standard deviation of the following set of arbitrary data. Write the data in  \mu \pm \sigma form. Are the data very precise?

1.01  1.00    0.86   0.93   0.95   
1.1   1.04    1.02   1.08   1.12
0.97  0.93    0.92   0.89   1.15

Which data points are most likely to be erroneous? How can you tell?

Example
Problem:

3. Solve the following equations for x using one of the rootfinding methods discussed earlier. Note that some equations have multiple real solutions (the number of solutions is written next to the equation)

a.  x^2 - 14x + 15 = 0 (2 solutions). Use the quadratic formula to check your technique before moving on to the next problems.

b.  x^2 - 14x + 15 - ln(x) = 0 (1 solution)

c.  e^{3x} = -x (1 solution)

d.  \frac{x}{2x^2 - 3} - \frac{2x^3-x^2}{2x-x^2} = 10 (2 solutions)