# Compressibility factor of hydrogen at low pressure and constant temperature

As we all know, the compressibility factor $$Z$$ of hydrogen and helium is always greater than $$1$$ at a constant moderate temperature. But, if we look at the virial equation of compressibility factor

$$Z = 1 + \frac{1}{V_\mathrm{m}}\left(a - \frac{b}{RT}\right) + \frac{b^2}{V_\mathrm{m}^2} + \cdots$$

At extremely low pressure $$V_\mathrm{m}$$ would be extremely large. So, the equation simplifies to

$$Z = 1 + \frac{1}{V_\mathrm{m}}\left(a - \frac{b}{RT}\right)$$

Now, the values of $$a$$ for hydrogen and helium are extremely small. So, we can take them as zero. The equation further simplifies to

$$Z = 1 - \frac{1}{V_\mathrm{m}}\left(\frac{b}{RT}\right)$$

Now, the value of $$b$$, $$R$$, $$T$$ and $$V_\mathrm{m}$$ are all positive. Though the value of $$b$$ for hydrogen is extremely small, still it is positive. So, shouldn't the value of $$Z$$ at some small pressure be less than $$1$$?

Any help would be appreciated.

• your value of $b/RT$ is also very small $\approx 0.0014$ so is smaller than $a$ so you should ignore this term and not $a$. – porphyrin Mar 3 at 11:20

## 1 Answer

In my textbook the opposite meaning is given to $$a$$ (an attractive parameter) and $$b$$ (a size parameter), but this may be a matter of differences in labelling, not in meaning. I will for the sake of consistency stick to your convention. For $$\ce{He}$$ the van der Waals parameters have the following values:

• $$b = \pu{3.4598 J mol-1 M-1}$$
• $$a = \pu{0.023733 M-1}$$

As you can see from the following figure at $$RT$$ ($$\pu{298K}$$), while your first approximation is quite accurate for $$\ce{He}$$ already at $$V_\mathrm m \approx \pu{1 L/mol}$$, your second approximation fails until higher $$V_\mathrm m$$. It is negative, but negliglibly so, and at that point is not particularly useful as an approximation as already $$Z \approx 1$$. The approximation fails (at RT) because the condition $$b/RT \gt a$$ is never observed: $$\frac{b}{RT} = \pu{0.0014 M-1}$$ whereas $$a = \pu{0.0237 M-1}$$

You can solve for the temperature at which these two become equal:

$$T_b = \frac{b}{aR}$$

For $$\ce{He}$$, $$T_b = 17.53\ \mathrm K$$. Below $$T_b$$ you may observe $$Z<1$$.

Sure enough, below that $$T$$ your second approximation starts to hold, as shown here for $$T = 10\ \mathrm K$$: or in terms of $$P$$: Note: it is not clear that the virial equation and van der Waals parameters should capture the behavior of the gases over a wide range of $$P$$ or $$V_m$$ at such low $$T$$.

• (1) Out of curiosity what temperature did you use? (2) for OP's assumption to hold it isn't sufficient that just $b/Rt \gt a$ but rather it must be that $b/Rt \gg a$. – MaxW Feb 24 at 0:14