In a continuous-flow reactor for methanol synthesis

CHEMISTRY · Multi-part · IB Diploma Programme · Chemistry · Ideal gases · Ideal gases

Problem

In a continuous-flow reactor for methanol synthesis, carbon monoxide must be metered accurately. The process engineer can operate the CO feed within the limits

T\le600\,\text{K}

and

P\le50\,\text{atm}

for safety reasons. Deviations from ideal-gas behavior must remain within

\pm5\%

(that is,

0.95\le Z\le1.05

) so that mass-flow controllers calibrated with the ideal-gas equation remain reliable. For CO, the van der Waals parameters are

a=1.50\,\text{L}^2\,\text{atm}\,\text{mol}^{-2}

and

b=0.0398\,\text{L}\,\text{mol}^{-1}

. (a) Derive an expression for the compressibility factor

Z

of a van der Waals gas in terms of

P

T

a

b

, and

R

. (b) Using the expression from part (a), calculate

Z

for CO at

T=600\,\text{K}

and

P=10\,\text{atm}

. State whether the deviation criterion is satisfied. (c) At

T=600\,\text{K}

, determine the maximum pressure (to the nearest atmosphere) that will keep

Z

within the acceptable range. Show your reasoning clearly. (d) Considering both safety limits and deviation limits, propose a single operating temperature–pressure pair for the CO feed that optimizes near-ideal behavior while respecting the constraints. Justify your recommendation quantitatively and discuss any trade-offs involved.

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