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Difficulty: MediumDeviations of Real Gases from Ideal Gas Behavior

A 1.5 mol1.5\text{ mol} sample of carbon dioxide gas (CO2\text{CO}_2) is held in a container of volume 0.60 dm30.60\text{ dm}^3 at a temperature of 300 K300\text{ K} under a pressure of 50 atm50\text{ atm}. What is the compressibility factor (ZZ) for the gas under these conditions? (Take R=0.082 atm dm3 mol−1 K−1R = 0.082\text{ atm dm}^3\text{ mol}^{-1}\text{ K}^{-1} and express your answer to two decimal places.)

Answer: 0.81

Answer

The compressibility factor (Z) for carbon dioxide under these conditions is 0.81.
The compressibility factor ZZ is defined as Z=PVnRTZ = \frac{PV}{nRT}. Substituting the given values yields PV=50×0.60=30.0PV = 50 \times 0.60 = 30.0 and nRT=1.5×0.082×300=36.9nRT = 1.5 \times 0.082 \times 300 = 36.9. Dividing 30.030.0 by 36.936.9 yields approximately 0.810.81, which indicates negative deviation from ideality due to intermolecular attractive forces.

Step-by-Step Solution

1
Identify the given variables and the formula for the compressibility factor.
P=50 atmP = 50\text{ atm}, V=0.60 dm3V = 0.60\text{ dm}^3, n=1.5 moln = 1.5\text{ mol}, T=300 KT = 300\text{ K}, R=0.082 atm dm3 mol−1 K−1R = 0.082\text{ atm dm}^3\text{ mol}^{-1}\text{ K}^{-1}. Formula: Z=PVnRTZ = \frac{PV}{nRT}.
The compressibility factor ZZ measures the deviation of a real gas from ideal gas behavior (Z=1Z = 1 for an ideal gas).
2
Calculate the actual pressure-volume product (PVPV).
PV=50 atm×0.60 dm3=30.0 atm dm3PV = 50\text{ atm} \times 0.60\text{ dm}^3 = 30.0\text{ atm dm}^3.
This represents the numerator in the compressibility ratio.
3
Calculate the theoretical ideal pressure-volume product (nRTnRT).
nRT=1.5 mol×0.082 atm dm3 mol−1 K−1×300 K=36.9 atm dm3nRT = 1.5\text{ mol} \times 0.082\text{ atm dm}^3\text{ mol}^{-1}\text{ K}^{-1} \times 300\text{ K} = 36.9\text{ atm dm}^3.
This represents the expected PVPV value if the gas behaved ideally.
4
Compute the value of ZZ and round to two decimal places.
Z=30.036.9≈0.8130→0.81Z = \frac{30.0}{36.9} \approx 0.8130 \rightarrow 0.81.
Dividing the observed PVPV by nRTnRT gives Z<1Z < 1, indicating that intermolecular attractive forces dominate under these conditions.

Key Concept

Compressibility factor Z of real gases
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Topics

Deviations of Real Gases from Ideal Gas Behavior

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