A previous Virtual Chemistry Experiment illustrated how to use a Reaction Table to determine an equilibrium constant. The virtual experiment involved allowing oxygen and sulfur dioxide gases to react to yield sulfur trioxide.
O_{2} (g) + 2 SO_{2} (g) 2 SO_{3} (g)
Perform the experiment several times using various starting pressures in the two bulbs. Does each experiment produce the same
equilibrium constant?
The equilibrium constant for this reaction is K_{P} = 124 atm^{-1}. How close are your calculated values to the
correct value? Why is there so much variability in the results for this experiment?
If all pressure measurements provided exactly correct values, then every trial for this experiment would give a value of precisely 124 atm^{-1} for K_{P}. The source of variability in K_{P} is variability and imprecision in the values for the initial and final pressures.
A manometer is employed for each pressure measurement. To obtain a single pressure, the heights of two columns of mercury must be read. The computer display has a limited resolution, with each pixel in the display corresponding with a height of 2.2 mm of Hg. Because the display is necessarily rounded to the nearest pixel, there is an uncertainty of ±1.1 mm in each height measurement. (Technically, the standard deviation is 0.6 torr.)
The uncertainty in the differences in the heights of the two columns of mercury is approximately ±1.6 mm of Hg or ±1.6 torr. (The standard deviation of each measured pressure is 0.9 torr.) In other words, even if one has perfect vision and correctly reads the height of the column of mercury to the correct pixel, a given pressure can never to known more precisely than approximately 1.6 torr. Thus there is an unavoidable random error in each pressure measurement owing to the limited precision of the instrument (in this case, the computer display). If one does not read the height of a column of Hg to the correct pixel, then there is an addition source of error.
Because the initial and final pressures are used to calculate K_{P}, the random error in the pressures produces a random error in K_{P}. How large is the random error in K_{P}?
Assessing how random error in one or more variables leads to random error in a calculated result is called a propagation of error calculation. There are mathematically formulas for performing a propagation of error calculation, but in this case the formulas are fairly complicated. An easier approach is trial and error. Let's begin by reviewing the equilibrium calculations.
The reaction table for this system is
O_{2} | SO_{2} | SO_{3} | |
---|---|---|---|
Initial | P_{1} | P_{2} | P_{3} |
Change | - x | - 2 x | + 2 x |
Equilibrium | P_{1} - x | P_{2} - 2 x | P_{3} + 2 x |
The three pressure measurements are for the initial pressure of oxygen in the left bulb (P_{L}), the initial pressure of sulfur dioxide in the
right bulb (P_{R}), and the final equilibrium pressure of the system (P). (Each of these pressures has an uncertainty of ±1.6 torr.)
The initial pressure of oxygen is P_{1} = ½ P_{L}
The initial pressure of sulfur dioxide is P_{2} = ½ P_{R}
The initial pressure of sulfur trioxide is P_{3} = 0
Thus the reaction table may be written as
O_{2} | SO_{2} | SO_{3} | |
---|---|---|---|
Initial | ½ P_{L} | ½ P_{R} | 0 |
Change | - x | - 2 x | + 2 x |
Equilibrium | ½ P_{L} - x | ½ P_{R} - 2 x | 2 x |
The extent of reaction, x, is obtained from the equilibrium and initial pressures:
P = P_{O2} + P_{SO2} + P_{SO3}
P = ½ P_{L} + ½ P_{R} - x
x = ½ P_{L} + ½ P_{R} - P
Once x has been calculated, the equilibrium constant may be found using
K_{P} = | P_{SO3}^{2} P_{O2} P_{SO2}^{2} |
= | 4 x^{2} (½ P_{L} - x) (½ P_{R} - 2 x)^{2} |
= | 32 x^{2} (P_{L} - 2 x) (P_{R} - 4 x)^{2} |
Evaluate how the calculated value of K_{P} is affected by random error in the measured pressures.
Part 1.Suppose one wished to determine K_{P} as accurately as possible using the virtual experiment.
How might one choose the experimental conditions to minimize the experimental error in K_{P}?
The working equations are
x = ½ P_{L} + ½ P_{R} - P
K_{P} = | P_{SO3}^{2} P_{O2} P_{SO2}^{2} |
= | 32 x^{2} (P_{L} - 2 x) (P_{R} - 4 x)^{2} |