Problem — Critical temperature from Van der Waals

Problem

For a gas, $a = 0{,}14$ Pa$\cdot$m$^6$/mol$^2$ and $b = 3{,}9\cdot10^{-5}$ m$^3$/mol. Estimate the critical temperature with the formula $T_c = 8a/(27Rb)$. Compare it with the real value for $\text{N}_2$ ($T_c \approx 126$ K).

Solution

I directly apply the formula for the critical temperature of the Van der Waals gas: $T_c = \frac{8a}{27\,R\,b}$ $T_c = \frac{8 \cdot 0{,}14}{27 \cdot 8{,}31 \cdot 3{,}9\cdot10^{-5}}$ $T_c = \frac{1{,}12}{8{,}75\cdot10^{-3}}$ $\ev{T_c \approx 128\ \text{K}}$ The estimated value is in excellent agreement with the real value for nitrogen ($T_c \approx 126$ K): the Van der Waals model captures well the behaviour near condensation.

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Topics: Kinetic theory of gases Concepts: Temperature Skills: Symbolic setup · Dimensional analysis