The given graph represents the variations of compressibility factor `Z=PV//nRT` vs `P` for three real gases `A`, `B`, and `C`.

Identify the incorrect statements.

Options

A. For gas `A`, `a=0` and its dependence on `p` is linear at all pressures.
B. For gas `B`, `b=0` and its dependence on `p` is linear at all pressures.
C. For gas `C`, which is a typical real gas, neither `a` nor `b=0`. By knowing the minima and power of intersection with `Z=1`, `a` and `b` can be calculated.
D. At high pressure, the slope is positive for all real gases.

Explanation

Compressibility factor,
`Z=(PV)/(nRT)`
At high value of pressure, volume occupied by the gas is very small, so `b` cannot be neglected, but due to a large value of `P`, `a//V^(2)` can be neglected in comparison to `P`. So, van der Waals equation for `1 mol` of the gas may be modified as follows:
`(P+(a)/(V^(2)))(V-b)=RT`
`(a)/(V^(2)) lt lt P`
`P(V-b)=RT`
or `PV-Pb=RT`
or `PV=RT+Pb`
`:. Z=(PV)/(RT)=(RT+Pb)/(RT)=1+(Pb)/(RT)`
This shows `Zgt1` and also it should be increasing linearly with pressure.
So, statement (`b`) should be false.
The value of `Z` generally increase with pressure and decreases with temperature. At higher pressures, molecules clooide more often, and at lower temperatures, they move less rapidly. This allows attractive forces between molecules to have a noticeable effect, making the volume of the real gas less than the volume of an ideal gas which causes `Z` to drop below `1`. When pressures are lower or temperatures are higher, the molecules are more free to move. In this case, the repulsive forces dominate, making `Zgt1`.