# Question #b1cd1

##### 1 Answer

Because gases really aren't always negligible in size compared to their container, particularly at high pressures and low temperatures.

Using

#barV_"real" = "2.1508 L/mol"# (Actual value)

#barV_"RK" = "2.1827 L/mol"# (Redlich-Kwong,#1.48%# error)

#barV_"vdW" = "2.2324 L/mol"# (van der Waals,#3.79%# error)

#barV_"id" = "2.3904 L/mol"# (Ideal Gas Law,#11.1%# error!!)

And it is clear that the ideal gas law overestimates the size of

Under these conditions, according to its phase diagram:

...

**VAN DER WAALS EQUATION OF STATE**

The **van der Waals** (vdW) *equation of state* is given by:

#P = (RT)/(barV - b) - a/(barV^2)# where:

#P# ,#V# ,#n# ,#R# , and#T# are known from the ideal gas law.#barV = V/n# is the molar volume.#a# is a vdW constant that accounts for the attractive and repulsive forces in the gas sample. Its units can be#"L"^2cdot"bar/mol"^2# if pressure is in#"bar"# .#b# is a vdW constant that accounts for the volume excluded by the gas particles (#b > 0# ). Its units are#"L/mol"# .

*This equation gives better agreement than the ideal gas law would, particularly at higher pressure and lower temperatures where the ideal gas law fails.*

**EXAMPLE: CO2**

Consider

#color(green)(barV_"real") = "1 L"/(20.4617 cancel"g") xx (44.009 cancel"g")/"mol" = ulcolor(green)("2.1508 L/mol")#

The ideal gas law would give:

#color(green)(barV_"id") -= V/n = (RT)/P#

#= (("0.083145 L"cdotcancel"bar""/mol"cdotcancel"K")(230 cancel"K"))/cancel"8 bar"#

#=# #ulcolor(green)("2.3904 L/mol")# which is fairly off, though it isn't bad. It means we used the equation in a physically valid situation.

With the vdW equation, however, we may get a better result. For

One would solve for

#PbarV^3 - (bP + RT)barV^2 + abarV - ab = 0#

In an iterative calculation (Newton-Raphson method), one could solve this equation to get three roots. Choosing the correct root would yield:

#color(blue)(barV_"vdW" = ul"2.2324 L/mol")#

It's not perfect, but it is definitely *closer* than the ideal gas law was.

Alternatively, the more complicated *Redlich-Kwong* equation of state, with

#barV_"real" = "2.1508 L/mol"#

#barV_"RK" = "2.1827 L/mol"#

#barV_"vdW" = "2.2324 L/mol"#

#barV_"id" = "2.3904 L/mol"#

**This illustrates the lack of ideality that a gas could exhibit at high pressure and low temperature.**