# Chemical Equilibria

## Uncertainty in Equilibrium Calculations

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.

O2 (g) + 2 SO2 (g) 2 SO3 (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 KP = 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?

### Experimental Error

If all pressure measurements provided exactly correct values, then every trial for this experiment would give a value of precisely 124 atm-1 for KP. The source of variability in KP 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.

### Propagation of Error

Because the initial and final pressures are used to calculate KP, the random error in the pressures produces a random error in KP. How large is the random error in KP?

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

O2 SO2 SO3
Initial P1 P2 P3
Change - x - 2 x + 2 x
Equilibrium P1 - x P2 - 2 x P3 + 2 x

The three pressure measurements are for the initial pressure of oxygen in the left bulb (PL), the initial pressure of sulfur dioxide in the right bulb (PR), 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 P1 = ½ PL

The initial pressure of sulfur dioxide is P2 = ½ PR

The initial pressure of sulfur trioxide is P3 = 0

Thus the reaction table may be written as

O2 SO2 SO3
Initial ½ PL ½ PR 0
Change - x - 2 x + 2 x
Equilibrium ½ PL - x ½ PR - 2 x 2 x

The extent of reaction, x, is obtained from the equilibrium and initial pressures:

P = PO2 + PSO2 + PSO3

P = ½ PL + ½ PR - x

x = ½ PL + ½ PR - P

Once x has been calculated, the equilibrium constant may be found using

 KP = PSO32PO2 PSO22 = 4 x2(½ PL - x) (½ PR - 2 x)2 = 32 x2(PL - 2 x) (PR - 4 x)2

### Experiment

Objective:

Evaluate how the calculated value of KP is affected by random error in the measured pressures.

Part 1.
1. Perform the experiment to determine KP, making note of the three measured pressures: PL (the pressure in the left flask before the gases are mixed), PR (the pressure in the right flask before the gases are mixed), and P, the final equilibrium pressure.

2. Enter PL and PR in the table below. Although these two pressures also have random error, for the purposes of this analysis, these pressures will be treated as exact (no random error). Based upon these pressures and the known equilibrium constant (KP = 124 atm-1), the true equilibrium pressure will be calculated and displayed.

How does the true P compare with the value you measured?

Enter your experimental value for P and the error will be displayed.
How does the random error compare with the value of ±1.6 torr discussed above? Are you able to read the heights of the columns of mercury to the best available precision?

How close is the experimental KP to the true value of 124 atm-1?

3. Enter the true P in the box for the experimental P. The calculated KP should read 124 atm-1. Now vary the value for the experimental P slightly (±1 or ±2 torr).

How sensitive is the calculated value of KP to the value of P?
How precisely must P be known in order to determine KP to within ±1%?
Is it possible to achieve this precision with this "experimental manometer"?

 PL = torr PR = torr true P = torr experimental P = torr error in P = torr

Partial Pressures (atm)
(based upon experimental P)
O2 SO2 SO3
Initial
Change
Equilibrium
 calculated KP = torr error in KP = torr

Part 2.

Suppose one wished to determine KP as accurately as possible using the virtual experiment.
How might one choose the experimental conditions to minimize the experimental error in KP?

The working equations are

x = ½ PL + ½ PR - P

 KP = PSO32PO2 PSO22 = 32 x2(PL - 2 x) (PR - 4 x)2

1. Carefully examine these equations and determine how the choices of PL and PR affect the precision in the resulting value of KP.

2. After you have formulated a few ideas, check the boxes below and see how your ideas compare with those listed below.

3. Repeat the experiment using conditions that are expected to give poor values for KP and using conditions that are expected to give good values for KP. Do the results confirm your error analysis?

Equilibrium Constant           Reaction Table           Le Châtelier's Principle