Dumas-style molar mass: is the answer right for the wrong reason?
A student in an SL class determines the molar mass of a volatile unknown liquid using a flask method. She adds about 1 cm³ of the liquid to a clean, dry 268 cm³ conical flask (volume found by filling it with water from a burette, ±1 cm³), covers the mouth with a pierced bung, and stands the flask in a water bath at 99.0 °C until no liquid is visible. She then removes the flask, allows it to cool, and reweighs it: the mass of liquid collected is 0.523 g. Atmospheric pressure that day is 98.7 kPa, recorded on a barometer reading ±0.5 kPa. Treating the vapour as ideal she calculates M = mRT/pV = 61.2 g mol⁻¹.
Reference data: propan-1-ol M = 60.1 g mol⁻¹ (bp 97 °C); propan-2-ol M = 60.1 g mol⁻¹ (bp 83 °C); ethanoic acid M = 60.1 g mol⁻¹ (bp 118 °C). The unknown boils at about 97 °C. While handling the hot flask she notices that some vapour escapes before the bung is fully pushed home. Her partner writes: 'M came out 1.8 % high, which the escaping vapour explains.'
1. Use the data to identify the liquid and state which single piece of evidence is decisive rather than merely consistent. Show the calculation and comment on whether the calculated M agrees with your identification within experimental uncertainty.
Read the worked answer
Calculation: M = mRT/pV = (0.523 × 8.314 × 372) / (98.7 × 10³ × 268 × 10⁻⁶) = 1617 / 26.45 = 61.1 g mol⁻¹, i.e. 61 g mol⁻¹ to two significant figures (the volume and pressure limit it to 3 s.f. at best).
Consistency is not identification. All three candidates have M = 60.1 g mol⁻¹, so the molar mass cannot separate them: a value of 61 is within 2 % of all of them. The decisive evidence is the boiling point, 97 °C, which matches propan-1-ol (97 °C) and rules out propan-2-ol (83 °C) and ethanoic acid (118 °C).
Uncertainty check: mass 0.001/0.523 = 0.19 %, volume 1/268 = 0.37 %, temperature 0.5/372 = 0.13 %, pressure 0.5/98.7 = 0.51 %. Combined random uncertainty ≈ 0.7 %, so M = 61.1 ± 0.4 g mol⁻¹. The reference value 60.1 lies just outside this range, so the 1.8 % deviation is too large to be random: a systematic error is present and must be identified before the result is trusted.
2. Evaluate the partner's explanation. Propose the systematic error that actually fits the sign of the discrepancy, and explain why the obvious candidate does not.
Read the worked answer
The partner has the sign backwards. The measured mass is the liquid found in the flask after cooling. If vapour escapes while the flask is hot, less vapour remains to condense, so the recorded mass is too small. Since M = mRT/pV, a mass that is too small gives a molar mass that is too low. The result is 1.8 % too high, so escaping vapour cannot be the explanation.
Errors that would give a value that is too high:
- Air not fully displaced. If air remains, the vapour's partial pressure is less than atmospheric, so n(vapour) < pV/RT while the full mass is still counted. Dividing the mass by too few moles inflates M. This is the most likely error, since the bung was pushed in early.
- The vapour was cooler than 372 K when it filled the flask (flask lifted out of the bath too soon), so the temperature used is too high and n is again underestimated.
- Non-ideal behaviour: at 99 °C, close to its boiling point, propan-1-ol vapour is extensively hydrogen-bonded, so Z < 1 and the real number of moles exceeds the ideal value, again inflating the calculated M.
A good answer distinguishes the direction of each error rather than simply listing possible mistakes.
3. Describe two changes to the procedure that would reduce the dominant error, explaining in each case how the change works.
Read the worked answer
Displace the air deliberately and seal at bath temperature. Add a slight excess of liquid (or use a long-necked flask) so that vaporisation sweeps air out through the pinhole, and push the bung fully home while the flask is still immersed in the 99.0 °C bath. This ensures that the gas in the flask is essentially pure vapour at atmospheric pressure, so n = pV/RT is valid and the systematic error is removed rather than merely reduced.
Measure the temperature of the vapour itself, not just the bath, using a thermometer inserted through the bung, and record it at the moment of sealing. The bath can be several degrees hotter than the flask contents, and T enters the calculation directly.
Supplementary improvements worth credit: use a larger sample and a larger flask to lower the relative uncertainty in volume; read the barometer at the time of the experiment; use a balance reading to ±0.0001 g; run the determination in triplicate and compare the spread of results with the propagated uncertainty, so that a systematic error is detectable.