Show that the van der Waals and Redlich-Kwong equations ofstate reduce to PV=nRT

Question

Show that the van der Waals and Redlich-Kwong equations ofstate reduce to PV=nRT in the limit of low density. van der Waals: P= RT/(Vm-b) -a/Vm2 Redlich-Kwong: P= RT/(Vm-b) - (a/T *1/Vm(Vm+b)) Note: Please show the process of thinking for the beginningsteps. I just do not know where to start or how to apply the limitof having a low density. Show that the van der Waals and Redlich-Kwong equations ofstate reduce to PV=nRT in the limit of low density. van der Waals: P= RT/(Vm-b) -a/Vm2 Redlich-Kwong: P= RT/(Vm-b) - (a/T *1/Vm(Vm+b)) Note: Please show the process of thinking for the beginningsteps. I just do not know where to start or how to apply the limitof having a low density.

Explanation / Answer

Vm = V/n = (n/V)-1 = -1 If density is low than p-1 is large => Vm islarge That means a/Vm => a/ => 0 Vm is large => Vm >> b   => Vm -b ~ Vm Thus P= RT/(Vm-b) - a/Vm2 =>RT/Vm - 0 = RT/Vm P = RT/Vm = nRT/V    , so   1/[Vm*(Vm+b)] => 1/=> 0 So P = RT/(Vm-b) - (a/T *1/Vm(Vm+b)) => RT/Vm - 0 =>RT/Vm

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