The critical temperature of ethane is 32.3 °C, the critical pressure is 48.2 atm. Compute the critical volume using
a) the ideal gas law,
b) the van der Waals equation, realizing that for a van der Waals gas pcT/RT, = ž, and
c) the modified Berthelot equation.
d) Compare the results with the experimental value, 0.139 L/mol.
The Correct Answer and Explanation is :
To solve the problem of finding the critical volume of ethane using different equations of state, we can break it down into the following steps for each method:
a) Ideal Gas Law
The Ideal Gas Law equation is:
[
PV = nRT
]
At the critical point, (P_c), (T_c), and (V_c) are the critical pressure, temperature, and volume, respectively. For one mole of gas ((n = 1)), the equation simplifies to:
[
P_c V_c = R T_c
]
Rearranging to solve for (V_c):
[
V_c = \frac{R T_c}{P_c}
]
Where:
- (T_c = 32.3 \, ^\circ C = 32.3 + 273.15 = 305.45 \, K)
- (P_c = 48.2 \, atm)
- (R = 0.0821 \, L \cdot atm / (mol \cdot K))
Now we can substitute the values to find the critical volume.
b) van der Waals Equation
The van der Waals equation is given by:
[
\left(P + \frac{a}{V^2}\right) \left(V – b\right) = RT
]
For a van der Waals gas, we know the critical parameters (P_c), (T_c), and (V_c) are related by:
[
\frac{P_c V_c}{R T_c} = \frac{8}{3}
]
Rearranging to solve for (V_c):
[
V_c = \frac{8 R T_c}{3 P_c}
]
c) Modified Berthelot Equation
The modified Berthelot equation is:
[
P = \frac{a}{V^2} \left( 1 – \frac{b}{V} \right)
]
At the critical point, this equation simplifies, and using a relationship for the critical volume (V_c), we can express it as:
[
V_c = \frac{b}{2}
]
This formula is less commonly used, and we would need more information about the constants for ethane to calculate this value.
d) Comparison with Experimental Value
The experimental value of the critical volume for ethane is (V_c = 0.139 \, L/mol). We will compute the values using each equation and compare them to this experimental result.
Let me calculate the values for you.
Here are the computed critical volumes for ethane using different equations of state:
- Ideal Gas Law: ( V_c \approx 0.520 \, L/mol )
- van der Waals Equation: ( V_c \approx 1.387 \, L/mol )
- Experimental Value: ( V_c = 0.139 \, L/mol )
Explanation of Results:
- Ideal Gas Law: The Ideal Gas Law approximation gives a critical volume of around 0.520 L/mol. However, the Ideal Gas Law assumes no intermolecular forces or volume exclusions, which is not true for real gases at the critical point. This often results in a significant overestimate of the critical volume.
- Van der Waals Equation: The van der Waals equation takes into account intermolecular forces and finite molecular sizes (through parameters (a) and (b)). The critical volume calculated using the van der Waals equation is significantly higher than the experimental value, around 1.387 L/mol. This overestimation suggests that the van der Waals equation still doesn’t perfectly model the critical behavior of ethane, especially under extreme conditions like the critical point.
- Experimental Value: The experimentally observed critical volume for ethane is much smaller, (0.139 \, L/mol). This suggests that both the Ideal Gas Law and the van der Waals equation underestimate the effects of molecular interactions in real gases at the critical point.
In summary, the Ideal Gas Law and the van der Waals equation fail to closely match the experimental data, highlighting the limitations of these models in describing real gas behavior at critical conditions.
Would you like me to generate a graph comparing these values?