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Question: The compressibility factor for a real gas at high pressure is ____. A. \(1 + \dfrac{{RT}}{{Pb}}\) ...

The compressibility factor for a real gas at high pressure is ____.
A. 1+RTPb1 + \dfrac{{RT}}{{Pb}}
B. 1
C. 1+PbRT1 + \dfrac{{Pb}}{{RT}}
D. PbRT\dfrac{{Pb}}{{RT}}

Explanation

Solution

The compressibility factor is a thermodynamic property for modifying the ideal gas equation to account for behavior of real gases and is denoted using Z. The compressibility can be found using the formula Z=PVRTZ = \dfrac{{PV}}{{RT}}. Find the compressibility factor using the real gas equation.

Complete answer: or Complete step by step answer:
We are given to find the compressibility factor for a real gas at high pressure.
We are using the real gas equation to find the compressibility factor.
Real gas equation is also known as van der Waals equation.
Real gas equation is (P+aV2)(V−b)=RT\left( {P + \dfrac{a}{{{V^2}}}} \right)\left( {V - b} \right) = RT, where P is the pressure, V is the volume, T is the temperature, R is the ideal gas constant, a is the constant for attraction between the molecules of the real gas and b is the volume taken up by those molecules.
When the pressure is high then the volume is less, because the volume of a given gas is inversely proportional to pressure at constant temperature.
This means, the volume is less and the particles collide with the walls more frequently and the attraction decreases.
Therefore the term aV2\dfrac{a}{{{V^2}}} will be negligible, almost equals to zero.
Then the real gas equation becomes
(P+0)(V−b)=RT ⇒PV−Pb=RT ⇒PV−PbRT=1 ⇒PVRT−PbRT=1 ⇒PVRT=1+PbRT  \left( {P + 0} \right)\left( {V - b} \right) = RT \\\ \Rightarrow PV - Pb = RT \\\ \Rightarrow \dfrac{{PV - Pb}}{{RT}} = 1 \\\ \Rightarrow \dfrac{{PV}}{{RT}} - \dfrac{{Pb}}{{RT}} = 1 \\\ \Rightarrow \dfrac{{PV}}{{RT}} = 1 + \dfrac{{Pb}}{{RT}} \\\
But we know that the compressibility factor is equal to Z=PVRTZ = \dfrac{{PV}}{{RT}}, substitute Z in the place of PV/RT in the above equation.
PVRT=1+PbRT ∵Z=PVRT ∴Z=1+PbRT  \dfrac{{PV}}{{RT}} = 1 + \dfrac{{Pb}}{{RT}} \\\ \because Z = \dfrac{{PV}}{{RT}} \\\ \therefore Z = 1 + \dfrac{{Pb}}{{RT}} \\\
Therefore, the compressibility factor Z for a real gas at high pressure is 1+PbRT1 + \dfrac{{Pb}}{{RT}}
The correct option is Option C, 1+PbRT1 + \dfrac{{Pb}}{{RT}}

So, the correct answer is “Option C”.

Note:
The compressibility factor is used as a correction factor to ideal behavior. Do not confuse compressibility factor with isothermal compressibility coefficient as the isothermal compressibility coefficient is the fractional change in volume when there is a change in pressure at constant temperature.