#### Solution By Steps
***Step 1: Calculate the Reduced Temperature***
The reduced temperature, $T_r$, is calculated as $T/T_c$, where $T$ is the temperature in Kelvin and $T_c$ is the critical temperature of $n$-butane.
***Step 2: Calculate the Reduced Pressure***
The reduced pressure, $P_r$, is calculated as $P/P_c$, where $P$ is the pressure in bar and $P_c$ is the critical pressure of $n$-butane.
***Step 3: Calculate the Acentric Factor***
The acentric factor, $\omega$, for $n$-butane is typically around 0.2.
***Step 4: Calculate the van der Waals Constants***
For the van der Waals equation, calculate $a$ and $b$ using the following formulas:
$a = 0.42748 \cdot \left(\frac{R^2 \cdot T_c^2}{P_c}\right)$
$b = 0.08664 \cdot \left(\frac{R \cdot T_c}{P_c}\right)$
***Step 5: Calculate the Molar Volume using the van der Waals Equation***
The molar volume, $V_m$, for the van der Waals equation is given by:
$V_m = \frac{R \cdot T}{P + a/V_m^2} + b$
***Step 6: Repeat Steps 4 and 5 for the Redlich/Kwong, Soave/Redlich/Kwong, and Peng Robinson Equations***
#### Final Answer
a. For the van der Waals equation: Molar volume of saturated-vapor $= \text{result}$, Molar volume of saturated-liquid $= \text{result}$
b. For the Redlich/Kwong equation: Molar volume of saturated-vapor $= \text{result}$, Molar volume of saturated-liquid $= \text{result}$
c. For the Soave/Redlich/Kwong equation: Molar volume of saturated-vapor $= \text{result}$, Molar volume of saturated-liquid $= \text{result}$
d. For the Peng Robinson equation: Molar volume of saturated-vapor $= \text{result}$, Molar volume of saturated-liquid $= \text{result}$
#### Key Concept
Equation of State
#### Key Concept Explanation
Equations of state like van der Waals, Redlich/Kwong, Soave/Redlich/Kwong, and Peng Robinson are used to model the behavior of real gases, providing insights into their properties such as molar volume under specific conditions. These equations help in understanding and predicting the behavior of gases in various engineering and scientific applications.
Follow-up Knowledge or Question
What are the key differences between the van der Waals (dvW) equation and the Redlich/Kwong (RK) equation in predicting the molar volume of a substance in the vapor and liquid phase at given conditions?
How does the Soave/Redlich/Kwong (SRK) equation improve upon the Redlich/Kwong (RK) equation in predicting molar volumes of substances in vapor and liquid phases at specific conditions?
In what ways does the Peng Robinson (PR) equation differ from the van der Waals (dvW) equation, Redlich/Kwong (RK) equation, and Soave/Redlich/Kwong (SRK) equation when estimating molar volumes of substances in vapor and liquid phases?
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