Question

a. Using the relationships derived in Example Problem 7.1 and the values of the critical constants for water from Table \(7.2,\) calculate values for the van der Waals parameters \(a, b,\) and \(R\) from \(z_{c}, T_{c}, P_{c},\) and \(V_{c^{*}} .\) Do your results agree with those in Tables 1.2 and \(7.4 ?\) b. Calculate the van der Waals parameters \(a\) and \(b\) using the critical constants for water and the correct value for \(R .\) Do these results agree with those in Tables 1.2 and \(7.4 ?\)

Short answer

The van der Waals parameters 'a', 'b', and 'R' can be calculated using the given critical constants and compression factor (Zc) for water as follows: \(b = \frac{V_c}{Z_c}\), \(a = \frac{(27/64) \times (Z_c \times Tc)^2}{Pc}\), and \(R = \frac{Pc}{Tc \times Zc}\). Comparing the obtained values with those in Tables 1.2 and 7.4 is necessary to check the agreement. For part b, 'a' and 'b' can be calculated using the critical constants and the exact value of 'R': \(b = \frac{R \times T_c}{8 \times P_c}\) and \(a = \frac{3 \times (R \times T_c)^2}{32 \times P_c}\). Comparing these values with those in Tables 1.2 and 7.4 verifies the agreement.

Step-by-step solution

Calculation of a, b and R using Zc, Tc, Pc, and Vc*.

Given the critical constants, Tc (critical temperature), Pc (critical pressure), Vc (critical volume) and Zc (compression factor), R, a, b can be determined as follows: - The coefficient \( b \) can be calculated by the formula \( b = \frac{V_c}{Z_c} \) - \( a \) can be calculated from the formula \( a = \frac{(27/64) \times (Z_c \times Tc)^2}{Pc} \) - \( R \) can be obtained by dividing Pc by the product of Tc and Zc The given values for Tc, Pc, Vc and Zc should be used in the above formulas to find the parameters 'a', 'b' and 'R'.

Check agreement with tables.

Compare the obtained 'a', 'b' and 'R' with the values listed in Tables 1.2 and 7.4. The second part of the exercise requests calculation of 'a' and 'b' using critical constants and the exact value for 'R'. Compare these obtained values with the provided tables.

Calculation of a and b using Tc, Pc, and correct R value.

'a' and 'b' can be calculated using the formulas given in step 1; however, in this case, the value of 'R' should be substituted with its exact value. - Compute \( b \) using the formula \( b = \frac{R \times T_c}{8 \times P_c} \) - Calculate \( a \) using the formula \( a = \frac{3 \times (R \times T_c)^2}{32 \times P_c} \)

Check agreement with tables' values.

Compare the calculated 'a', and 'b' with the corresponding values listed in Tables 1.2 and 7.4. By following these steps, it is possible to compute and verify the van der Waals parameters using the given critical constants and the correct value for the gas constant.

Key concepts

Critical Constants

Critical constants play an invaluable role in understanding the behavior of gases under extreme conditions. The critical temperature (\( T_c \)), critical pressure (\( P_c \)), and critical volume (\( V_c \)) are essential points on a phase diagram where a substance transitions between its gas and liquid states without a distinct phase change. Essentially, at these 'critical' values, the distinction between gas and liquid ceases to exist, and the substance is in a unique state known as a supercritical fluid.

Understanding these constants is particularly important when working with the van der Waals equation, as they can be used to calculate the parameters of real gases. Further, the critical compressibility factor (\( Z_c \)), which is typically less than 1 for real gases, offers insight into how much the gas deviates from ideal gas behavior at the critical point. It's a crucial factor in fine-tuning the van der Waals constants for specific substances.

van der Waals Equation

The van der Waals equation is a modified version of the ideal gas law that accounts for the finite size of molecules and the intermolecular forces between them. This is crucial as the ideal gas law assumes point-like particles with no attraction or repulsion, which isn't a true representation of real gases. The van der Waals equation introduces two parameters, \(a\) and \(b\), to correct these assumptions. The \(b\) parameter accounts for the volume occupied by the gas molecules, while \(a\) factors in the attractions between them.

To derive these parameters for a specific gas, we turn to the critical constants. For instance, from the critical volume, we can determine \(b\) as it's approximately equal to the volume occupied by one mole of gas molecules at the critical point. Similarly, \(a\) can be calculated using values of critical temperature and critical pressure to reflect the strength of intermolecular forces at the critical point. By adjusting the values of \(a\) and \(b\) accordingly, the van der Waals equation becomes a powerful tool for describing the properties of real gases.

Compressibility Factor

The compressibility factor (\(Z\)) is a dimensionless quantity that describes how much a gas deviates from ideal gas behavior. It's defined as the ratio of the product of pressure and volume to the product of the gas constant (\(R\)) and temperature. An ideal gas has a compressibility factor of 1, meaning it follows the ideal gas law perfectly. However, real gases have compressibility factors different from unity, indicating varying degrees of deviation, usually due to molecular interactions and finite molecular size.

At the critical point, the compressibility factor can yield values lower than 1, revealing the impact of strong intermolecular forces. When calculating the van der Waals parameters, the critical compressibility factor (\(Z_c\)) is often used. If the calculated parameters are accurate, they will correctly predict the gas's behavior at its critical point, where the deviations from ideal behavior are significant and cannot be ignored. Evaluating the compressibility factor alongside the van der Waals parameters provides a comprehensive understanding of a gas's behavior not just under typical conditions, but also in the vicinity of its critical point.