Question

The density of water vapour at $327.6atm$ and $776.4K$ is $133.2gm/d{m^3}$. Determine the molar volume, ${V_m}$ of water and the compression factor.
A. ${V_m} = 5.0775mol/L,Z = 1.457$
B. ${V_m} = 6.0775mol/L,Z = 13.457$
C. ${V_m} = 5.0775mol/L,Z = 4.457$
D. ${V_m} = 6.0775mol/L,Z = 1.457$

Answer 1

The molar volume of the substance is the volume occupied by the substance at a definite temperature and pressure. The compression factor is the correction factor which describes the deviation of a gas from ideal behavior

Complete step by step answer:
Molar volume is the ratio of molar mass of the substance and the density. It is expressed as,
${V_m} = \dfrac{M}{\rho }$, where M is the molar mass of the substance and $\rho$ is the density.
Given, density of water vapour = $133.2gm/d{m^3}$
Pressure = $327.6atm$
Temperature =$776.4K$
Following the combined gas laws for ideal gas the equation followed is expressed as,
$PV = nRT$
Where $P$is the pressure,
$V$ is the volume of the gas,
$n$ is the moles of gas occupied in that volume,
$R$is the gas constant,
$T$is the temperature,
The molar volume of the gas is the ratio of moles and the volume occupied.
Molar volume = $\dfrac{n}{V} = \dfrac{P}{{RT}}$
Inserting the values,
$\dfrac{n}{V} = \dfrac{{327.6}}{{0.0821 \times 776.4}}$
${V_m} = \dfrac{n}{V} = 5.14mol/L$
The molar mass of the water =$18g$.
Thus $1$mole of water contains $18g$of the substance.
The mole contained in $133.2g$ of water is equal to = $\dfrac{{133.2}}{{18}} = 7.4mole$
The molar volume of given gas was calculated as = $5.14{\text{ }}mol/L$
The compressibility factor is the ratio of the mole of the gas and the molar volume.
Thus $Z = \dfrac{n}{{{V_m}}}$
$Z = \dfrac{{7.4}}{{5.14}}$
$Z = 1.44$.
Hence the molar volume and the compressibility factor are $5.14mol/L$ and $1.44$

So, the correct answer is Option A.

Note: Molar volume is defined as the volume occupied by one mole of any gas at room temperature and pressure. Compression factor is also known as gas deviation factor. It shows the deviation of a real gas from ideal behavior. It could be a valuable thermodynamic property for altering the perfect gas law to account for the genuine gas behavior.