Solve the practice problem in the lecture notes.
The Redlich/Kwong Equation of State is $$ P = \frac{RT}{V-b} - \frac{a}{T^{1/2}V(V+b)}, $$ where $T$ is temperature, $V$ is molar volume, $R$ is the universal gas constant, $a$ and $b$ are compound-specific constants. Find the molar volume of ethane for the vapor phase that is present at $T=77$ $^o$C and $P=1$ bar. (Note, you should convert T to an absolute temperature scale (K).
For ethane, $a=2.877\times 10^8$ cm$^6$bar K$^{0.5}$mol$^{-2}$ and $b=60.211$ cm$^3$mol$^{-1}$.
Hints:
Find one solution for $x$ and $y$ to the following equations using Excel's Solver (there are 4 possible solutions). $$2x^2 + y^2 = 1 $$ $$(0.5x-0.5)^2 + 2(y-0.25)^2 = 1 $$
You may want to graph the equations to see where they intersect and find a good guess for the solver. If you do make a plot, note that for the first equation you can only solve $y$ for $x$ values between -0.7 and 0.7. For the second equation, you can only solve for $y$ with $x$ values between -1 and 3. Also, remember that when solving $y^2=x$ for $y$, the solution is $y=\pm\sqrt{x}$.
Imagine mixing liquid benzene (species 1) and toluene (species 2) together in an initially empty container. At equilibrium, some of the liquid from both species will evaporate into the vapor phase and some will be left as liquid for certain temperatures and pressures. In thermodynamics, Raoult's law may be used to describe the distribution of species in each phase. For the specific example mentioned above, Raoult's law gives the following two expressions describing the equilibrium state, $$y_1P = x_1P_1^{sat}(T), $$ $$y_2P = x_2P_2^{sat}(T), $$
where $y_i$ is the mole fraction of species $i$ (either 1 or 2) in the vapor phase, and $x_i$ is the mole fraction of species $i$ in the liquid phase, $P_i^{sat}(T)$ is the vapor pressure of species $i$ at the system temperature (T), and $P$ is the system pressure.
The vapor pressures can be found from the Antoine Equation, $$ \ln(P_i^{sat}(T)) = A_i - \frac{B_i}{T + C_i}.$$
Here, $A_i$, $B_i$, and $C_i$ are constants. In this equation, $T$ is evaluated in $^o$C, and $P_i^{sat}(T)$ is in kPa.
Compound | A | B | C |
---|---|---|---|
Benzene | 13.7819 | 2726.81 | 217.572 |
Toluene | 13.9320 | 3056.96 | 217.625 |
If our mixture has $y_1=0.33$ and $P=120$ kPa, find $x_1$ and T. Remember that the mole fractions of each phase must sum to 1.
Hints: