# B Negative deviation in PV vs P graph

1. Mar 3, 2017

### Abhi9826

I just wonder that why the negative deviation comes in graph except for hydrogen and helium gas.I am still thinking for the reason and searching for it.So please help me ...

2. Mar 3, 2017

You may see some effects as shown in your graph, but I believe the deviations that are shown are greatly exaggerated. I'd be curious to know what the source of the graph is, and if these substances do, in fact, show this behavior. $\\$ Editing...Presumably the curve is for constant temperature, and what it is showing is basically a huge drop in the expected volume as the pressure is increased, but then as the pressure is increased further, the volume is actually larger than expected from the ideal gas law. The Van der Waals equation ($(P+an^2/V^2)(V-nb)=nRT$), as opposed to the ideal gas equation, can serve to account for slight differences from the ideal gas law, and the forces accounting for the variations in the pressure part of the equation are often referred to as van der Waals forces. Perhaps if you solve for $PV$ in this equation and graph it vs. $P$, (for constant $T$), if you have a significant $a$ parameter from the molecule, you might get behavior as shown in your graph, but certainly not to the extent that is shown. $\\$ Additional editing: A little algebra on the Van der Waals equation gives $PV=nRT-an^2/V +Pnb+an^3 b/V^2$ and since $PV$ is approximately equal to $nRT$, we can call $1/V=CP$ for some constant $C$ (to a good approximation). Let $nRT=G$ for some constant $G$, and we have $PV=G-an^2CP+nbP+an^3 b C^2 P^2$ with $n$ a constant and $a$ and $b$ also constant for a substance. If $a$ is large enough, the $-an^2CP$ term could be larger than the $nbP$ term and could dominate the deviation of $PV$ from ideality (of being equal to $G$) before the $an^3 b C^2 P^2$ term takes over for large $P$. Perhaps this is what the graph you have is attempting to illustrate. For Helium, the attractive forces between molecules (which is what the $a$ term represents) are very small, (so that $a$ is very small), and any deviation from ideality is dominated by the $nbP$ term. The $b$ term in the Van der Waals equation results from atoms or molecules having a finite size. With a very small $a$, the $an^3 b C^2P^2$ term most likely is small for Helium for large $P$. Alternatively, for $CH_4$, the attractive forces between molecules is apparently quite significant, accounting for a sizable $a$ parameter, but certainly not to the extent depicted by the graph. $\\$ One additional comment: For complete mathematical rigor, because we ultimately looked at the $P^2$ term in the analysis, we should really write $1/V=CP+DP^2+...$ in an iterative solution for $1/V$. In a slightly simplified analysis, we assumed $D$ was negligibly small, but with complete mathematical rigor, we should have kept the $DP^2$ term in the algebra that followed.

Last edited: Mar 3, 2017
3. Mar 4, 2017

And one additional detail: A complete analysis of the $1/V$ expansion shows there may even be a constant term in what is basically a Taylor expansion type solution for $1/V=A+CP+DP^2+...$. The constant term $A$, along with the coefficient $D$ of the $P^2$ term doesn't appreciably affect the qualitative analysis that shows for a sizable $a$ that, in the graph of $PV$ vs. $P$, $PV$ will first take a dip below the $nRT$ value before going upward from the $P^2$ term as $P$ becomes larger.

4. Mar 4, 2017

### John Park

To provide a different sort of answer: the deviations from linearity are to do with the gases liquifiying. Hydrogen and helium don't become liquids until the temperature is very low, and may not show that behaviour at the temperatures represented in the plot.

The product PV is closely related to the so-called compressibility factor z = PV/nT. Its behaviour was (is?) an important topic in studies of the macroscopic properties of fluids.

In fact if volumes, temperatures, and pressures are scaled to the corresponding "critical" values (i.e. values for the state where liquid and gas are indistinguishable), most gases show very similar compressibility factors, over reasonable ranges of pressure. Look up the Law (or Theorem) of Corresponding States for more details.

Edit: Wikipedia has a good article on compressibility factors. The behaviours of hydrogen and helium are influenced by quantum effects.

5. Mar 4, 2017