Syringe compression: leak, dead volume or a lagging thermometer?
A student investigates the compression of air in a sealed transparent syringe. The syringe contains 20.0 cm³ of air at 21 °C and 101 kPa. The student quickly pushes the plunger to 8.0 cm³ in 0.6 s. A digital pressure sensor reads 240 kPa, and a thermocouple inside the gas reads 38 °C. The syringe barrel flexes slightly under pressure, and the pressure sensor has a manufacturer's tolerance of ±5%. The student's partner says the gas must have leaked because the pressure is lower than expected for an ideal gas at the measured temperature. The student is not sure: the compression was fast, so the temperature might be higher than the thermocouple reads because the thermocouple has a 0.4 s response time. The lab is at 101 kPa and 21 °C. The student wants to decide whether to repeat the experiment with a slower compression or to test for a leak.
1. Calculate the amount of air initially and the pressure predicted by the ideal gas law at 8.0 cm³ and 38 °C if no gas escapes. Compare with the sensor reading and decide whether a leak is proven.
Read the worked answer
Initial amount: n = pV/(RT) = (101 × 10³ Pa)(20.0 × 10⁻⁶ m³) / [(8.31 J mol⁻¹ K⁻¹)(294 K)] = 8.27 × 10⁻⁴ mol.
Predicted pressure at 8.0 cm³ and 311 K: p = nRT/V = (8.27 × 10⁻⁴)(8.31)(311) / (8.0 × 10⁻⁶) = 2.67 × 10⁵ Pa = 267 kPa.
The measured value is 240 kPa, about 10% lower. A 5% sensor tolerance is about ±12 kPa, so the difference is larger than the stated tolerance if the temperature reading is correct. However, that does not prove a leak. The thermocouple response time is 0.4 s, comparable with the 0.6 s compression, so the true gas temperature could be higher than 38 °C; a higher T would predict an even higher pressure, making the discrepancy worse, not better. A more likely systematic effect is dead volume in the sensor and connecting tube: the syringe scale may read 8.0 cm³ while the total gas volume is larger, which lowers the measured pressure. Barrel flexing also changes the volume.
A leak is therefore possible but not proven. Using the measured p, V and T gives n = (240 × 10³)(8.0 × 10⁻⁶)/[(8.31)(311)] = 7.43 × 10⁻⁴ mol, about 10% less than initially, but this assumes the volume is exactly 8.0 cm³ and the temperature is exactly 311 K.
2. Estimate the temperature rise expected for a reversible adiabatic compression from 20.0 cm³ to 8.0 cm³, starting at 21 °C, taking γ = 1.40. Compare with the measured 38 °C and explain the difference in terms of the first law.
Read the worked answer
For a reversible adiabatic compression, TV^(γ−1) = constant, so T₂ = T₁(V₁/V₂)^(γ−1) = 294 K × (20.0/8.0)^0.40 = 294 × 2.5^0.40.
2.5^0.40 = exp(0.40 ln 2.5) = exp(0.40 × 0.916) = 1.44, so T₂ ≈ 424 K, or about 151 °C.
The measured value is 38 °C (311 K), far below 151 °C. The process is therefore not adiabatic. In the first law, ΔU = Q + W, with W done on the gas. If the compression were adiabatic, all the work would raise the internal energy and the temperature would rise strongly. In reality, heat leaves the gas to the syringe walls and surroundings during the 0.6 s compression, so Q is negative and the temperature rise is much smaller. The compression is fast but not fast enough to be adiabatic.
3. The student repeats with a slower compression and finds the pressure at 8.0 cm³ is 255 kPa while the temperature returns to 21 °C. Use this to estimate whether the syringe has a leak, assuming the gas remains ideal.
Read the worked answer
At constant temperature, p₁V₁ = p₂V₂, so the expected pressure is p₂ = p₁V₁/V₂ = 101 kPa × 20.0/8.0 = 252.5 kPa.
The measured value is 255 kPa, about 1% higher. This is well within the pressure sensor tolerance and is consistent with no significant leak. The small excess could come from dead volume, barrel flexing or a slight temperature difference. The earlier 240 kPa reading was therefore probably affected by the non-equilibrium temperature distribution and the thermocouple lag, not by gas escaping. A definitive leak test would seal the syringe at a fixed volume and monitor pressure over several minutes; a steady fall would indicate a leak, while a constant reading would confirm sealing.