Question

Ethane in a rigid vessel is to be heated from 50 psia and \(100^{\circ} \mathrm{F}\) until its temperature is \(540^{\circ} \mathrm{F}\). What is the final pressure of the ethane as predicted by the compressibility chart?

Short answer

Based on the compressibility chart, the final pressure of ethane after being heated from an initial temperature of 100°F and pressure of 50 psia to a final temperature of 540°F is approximately 163 psia.

Step-by-step solution

Find the reduced values

First, we need to find the reduced temperature and reduced pressure for the initial conditions. The critical temperature (\(T_c\)) and critical pressure (\(P_c\)) of ethane can be found in a thermodynamics textbook or online. For ethane, \(T_c = 549.67 \mathrm{K}\) and \(P_c = 48.2 \mathrm{atm}\). Let's first convert the given temperatures and pressure to absolute values and the pressure to atmospheres. \(T_1 = 100^{\circ} \mathrm{F} + 459.67 = 559.67 \mathrm{R}\) and \(P_1 = \frac{50 \, \mathrm{psia}}{14.696} = 3.402 \mathrm{atm}\). Now, convert \(T_1\) to Kelvin: \(T_1 = \frac{5}{9}(559.67) = 310.93 \mathrm{K}\). Next, we will find the reduced temperature (\(T_{r1}\)) and reduced pressure (\(P_{r1}\)) by dividing the initial temperature and pressure by their respective critical values: \(T_{r1} = \frac{T_1}{T_c} = \frac{310.93 \:\mathrm{K}}{549.67 \:\mathrm{K}} = 0.565 \, (dimensionless)\), and \(P_{r1} = \frac{P_1}{P_c} = \frac{3.402 \: \mathrm{atm}}{48.2 \: \mathrm{atm}} = 0.071 \, (dimensionless)\).

Find the initial compressibility factor, \(Z_1\)

Using the reduced temperature and pressure we calculated in step 1, we can now find the initial compressibility factor (\(Z_1\)) using the compressibility chart. The values on the chart are approximations, so it is normal to have some uncertainty. By locating the intersection of the isotherm curve for \(T_{r1} = 0.565\) and the isobar for \(P_{r1} = 0.071\), we approximate that \(Z_1 \approx 0.85\).

Find the initial molar volume, \(V_1\)

Now, we will use the ideal gas law and the calculated value of \(Z_1\) to find the initial molar volume, \(V_1\): \(PV = ZnRT\); thus, \(V_1 = \frac{Z_1 RT_1}{P_1} = \frac{0.85 \times 0.0821 \times 310.93 \:\mathrm{K}}{3.402 \: \mathrm{atm}} = 6.916 \: \mathrm{L/mol}\).

Determine the final reduced volume and final reduced temperature

Since it is a rigid vessel, the molar volume remains constant as the ethane is heated. Therefore, \(V_2 = V_1 = 6.916 \mathrm{L/mol}\). Now, we can calculate the final reduced volume (\(V_{r2}\)) by dividing the final molar volume (\(V_2\)) by the critical molar volume of ethane (\(V_c = \frac{RT_c}{P_c}\)). \(V_c = \frac{0.0821 \times 549.67 \:\mathrm{K}}{48.2 \: \mathrm{atm}} = 0.0938 \: \mathrm{L/mol}\), and \(V_{r2} = \frac{V_2}{V_c} = \frac{6.916 \: \mathrm{L/mol}}{0.0938 \: \mathrm{L/mol}} = 73.72\, (dimensionless)\). Now, we need to find the final reduced temperature (\(T_{r2}\)). Convert the final temperature to absolute values and Kelvin: \(T_2 = 540^{\circ} \mathrm{F} + 459.67 = 999.67 \mathrm{R}\), and \(T_2 = \frac{5}{9}(999.67) = 555.37 \mathrm{K}\). We can then find \(T_{r2}\): \(T_{r2} = \frac{T_2}{T_c} = \frac{555.37 \: \mathrm{K}}{549.67 \: \mathrm{K}} = 1.01 \, (dimensionless)\).

Find the final compressibility factor, \(Z_2\)

With the final reduced volume and final reduced temperature, we can find the final compressibility factor (\(Z_2\)) using the compressibility chart. Locating the intersection of the isotherm curve for \(T_{r2} = 1.01\) and the isochore for \(V_{r2} = 73.72\), we estimate \(Z_2 \approx 1.15\).

Calculate the final pressure

Finally, we can calculate the final pressure (\(P_2\)) using the ideal gas law and the final compressibility factor (\(Z_2\)): \(P_2 = \frac{Z_2 RT_2}{V_2} = \frac{1.15 \times 0.0821 \times 555.37 \: \mathrm{K}}{6.916 \: \mathrm{L/mol}} = 11.09 \: \mathrm{atm}\). Since we need the final pressure in psia, we can convert it back to psia: \(P_2 = 11.09 \times 14.696 = 163 \: \mathrm{psia}\) (approximately). So, the final pressure of the ethane, as predicted by the compressibility chart, is approximately 163 psia.