Compress the Headspace: the Ideal Gas Law

The gas in the headspace above the broth in a sealed bioreactor behaves as an ideal gas. Its pressure depends on the amount of gas, its temperature and the volume it occupies. First change one thing at a time and see how the pressure responds. Then use the full ideal gas law, including the unit conversions, and finally test your knowledge. Work through the three stages in order.

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Choose what to change:

Volume 0.50 m³

Temperature 37 °C

Amount of gas 43 mol

New pressure kPa

Answers within 5% are accepted, so small rounding differences do not matter.

Remember

PV = nRT. For a fixed amount of gas, P₁V₁/T₁ = P₂V₂/T₂.
Temperature must be in kelvin: K = °C + 273.15. The pressure of an ideal gas would fall to zero at 0 K (−273.15 °C), so pressure is proportional to the kelvin temperature, not the Celsius temperature.
With R = 8.314 J mol⁻¹ K⁻¹, use P in Pa and V in m³. 1 m³ = 1,000 L; 1 kPa = 1,000 Pa; 1 bar = 100 kPa; 1 atm = 101.325 kPa.
The vessel in this lesson is rated to 500 kPa, and the gauge reads absolute pressure.
This is an ideal gas: it ignores the size of the molecules and the forces between them, which is a good approximation for air and CO₂ in a headspace.