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Question: The compressibility factor of \({ CO }_{ 2 }\) at 273 K and 100 atm pressure is 0.2005. The volume o...

The compressibility factor of CO2{ CO }_{ 2 } at 273 K and 100 atm pressure is 0.2005. The volume occupied by 0.2 mole of CO2{ CO }_{ 2 } gas at 100 atm and 273 K using real gas nature is 8.89×10−x8.89\times { 10 }^{ -x } Litre.
So the value of x is…

Explanation

Solution

The compressibility factor gives us an idea about the degree to which a real gas shows deviation from the ideal gas behaviour. It is the ratio of the observed molar volume of a gas to the calculated molar volume (using ideal gas equation) of the gas at the same pressure and temperature

Complete answer:
The compressibility factor for a gas is the ratio of the observed molar volume of a gas to the calculated molar volume (using ideal gas equation) of the gas at the same pressure and temperature. It can also be defined as the ratio of the product and pressure and volume of the gas to the product of the number of moles, the gas constant and the temperature of the gas. The equations are given below:
Z=PVnRTZ=\cfrac { PV }{ nRT } (Where Z is the compressibility factor)
⇒Z=PVrealnRT\Rightarrow Z=\cfrac { P{ V }_{ real } }{ nRT }...(1)
Since according to the ideal gas equation: PV=nRT PV=nRT
⇒Vreal=nRTP\Rightarrow { V }_{ real }=\cfrac { nRT }{ P }
Putting the above equation in equation (1),
Z=VrealVidealZ=\cfrac { { V }_{ real } }{ { V }_{ ideal } }
We will use equation (1) to solve this question. Putting the value of n=0.2 mole, Z=0.2005, P=100 atm, R=0.0821 L atm K−1mol−1{ K }^{ -1 }{ mol }^{ -1 } and T=273 K, we get:
0.2005=1000atm×V0.2mol×0.0821LatmK−1mol−1×273K0.2005=\cfrac { 1000\quad atm\times V }{ 0.2\quad mol\times 0.0821\quad L\quad atm\quad { K }^{ -1 }{ mol }^{ -1 }\times 273\quad K }
⇒V=8.89×10−3L\Rightarrow V=8.89\times { 10 }^{ -3 }\quad L
Hence the value of x is 3.

Note: The compressibility factor for an ideal gas is 1 while for a real gas it could be less than 1 or greater than 1. If the compressibility factor is less than 1 then it implies that the attractive forces are predominant among the gas molecules. If the compressibility factor is more than 1 it implies that the repulsive forces are predominant among the gas molecules. Whether the compressibility factor will be greater than or less than 1 at a particular temperature and pressure will depend upon the nature of the gas.