Question

At low pressure, the van der Waal's equation is reduced to

• A. $Z=\frac{PV_{m}}{RT}=1-\frac{a}{VRT}$
• B. $Z=\frac{PV_{m}}{RT}=1+\frac{b}{RT}P$
• C. $PV_{m}=RT$
• D. $Z=\frac{PV_{m}}{RT}=1-\frac{a}{RT}$

Answer 1

The Correct Option is A

When pressure is low
$\left[P+\frac{a}{V^{2}}\right]\left(V-b\right)=RT$
or $PV=RT+Pb-\frac{a}{V}+\frac{ab}{V^{2}}$
or $\frac{PV}{RT}=1-\frac{a}{VRT}$
$Z=-\frac{a}{VRT}$
$\left(\because\frac{PV}{RT}-Z\right)$

Answer 2

For ideal gases, Z=1 as PV = nRT.

Although, for real gasses, we may apply the Van der Waals equation i.e.

\([\frac{P+an^2}{V^2}]\) [V-nb] = nRT

Where,

P stands for pressure,

V  means volume,

n is the number of moles

a is pressure correction constant

b is volume correction constant

T stands for temperature

R is for the universal gas constant

At the lower pressure, the volume correction constant or b becomes almost 0 or negligible and the equation is, therefore,

(P+\(\frac{a}{v^2}\))V = R.T

⇒\(\frac{PV}{RT}\)=(1-\(\frac{a}{RTV}\))=Z

Here,

Z is the compressibility factor.

Therefore, from the above calculation, we understand that Z = \(\frac{PvM}{RT}\) = 1-\(\frac{a}{RTV}\) and the correct answer is option (A).