Ideal Gases — Question 10

Question 10

Differentiating the gas law separates filling from heating

Treat the gas as ideal, $pV=nRT$, and use $R=8.314\,\mathrm{J/(mol\,K)}$. A rigid .0100 m$^3$ tank is instantaneously at $p=200$ kPa and $T=300$ K. Gas enters while the tank temperature rises at 2.00 K/s. Pressure rises at 3.00 kPa/s. Treat these as instantaneous uniform-state rates.

1. Derive $\dot n$ from the gas law and calculate the molar inflow rate.

2. Find the apparent inflow rate if temperature change were ignored, and the threshold $\dot p$ corresponding to zero inflow at the stated $\dot T$.

Question 10 - Solution

Differentiating the gas law separates filling from heating

(a) Method and calculation

$$n=\frac{pV}{RT}\ \Rightarrow\ \dot n=\frac V{RT}\left(\dot p-\frac pT\dot T\right).$$$$\dot n=\frac{.0100}{8.314(300)}\left(3000-\frac{200000}{300}(2)\right)\approx.00668217\,\mathrm{mol/s}.$$

(b) Interpretation and further calculation

$$\dot n_{\mathrm{isothermal}}=\frac{.0100(3000)}{8.314(300)}\approx.0120279\,\mathrm{mol/s}.$$$$\dot n=0\ \Longleftrightarrow\ \dot p=\frac pT\dot T=1333.33\,\mathrm{Pa/s}.$$Only the pressure-rise portion above this thermal contribution indicates inflow; the isothermal estimate is 1.8 times the correct rate.

Physical check

An increasing pressure does not alone prove that molecules are entering.