Consider a liquid at 90 bar and 40 °C. What is the change in entropy of the substance as you change the conditions to 1 bar and 150 °C by expanding the liquid while heating?
Vapor pressure = 59.2 bar
heat of vaporization = 1.82 kJ/mol , both at 150 degree C
Heat capacity = 17.5 J/mol K
equation of pressure - PV/RT = 1 + (BP/RT)
2nd virial coeff, B = b - (a/T^2)
where a = 4.9*10^7 K^2cm^3/mol and b = 24cm^3/mol
To calculate the change in entropy of the substance as you change the conditions from 90 bar and 40 °C to 1 bar and 150 °C by expanding the liquid while heating, we can use the principles of thermodynamics and the given equations.
First, let's break down the problem into steps and calculate the entropy change at each step.
Step 1: Expanding the liquid from 90 bar to the vapor pressure of 59.2 bar
To find the change in entropy during this step, we can use the ideal gas equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
Since we know the initial pressure (90 bar), final pressure (vapor pressure of 59.2 bar), and final temperature (40 °C), we can calculate the initial and final volumes using the ideal gas equation.
For the initial conditions:
P1 = 90 bar
V1 = unknown
T1 = 40 °C + 273.15 = 313.15 K (convert to Kelvin)
For the final conditions:
P2 = 59.2 bar
V2 = unknown
T2 = 40 °C + 273.15 = 313.15 K (convert to Kelvin)
Now, let's rearrange the ideal gas equation to solve for the initial and final volumes:
V1 = (nRT1) / P1
V2 = (nRT2) / P2
Next, we need to calculate the number of moles (n) using the given information. Since we don't have the mass or volume of the liquid, we cannot directly calculate the number of moles. However, we can assume that the liquid is an ideal solution and use the van't Hoff equation to estimate the change in the logarithm of the vapor pressure with temperature:
ln(P2/P1) = ΔHvap / R * (1/T2 - 1/T1)
Where ΔHvap is the heat of vaporization, R is the ideal gas constant, and T1 and T2 are the initial and final temperatures.
Rearrange the equation to solve for ΔHvap:
ΔHvap = R * ln(P2/P1) / (1/T2 - 1/T1)
Substituting the given values into the equation, we can calculate ΔHvap.
Now that we have ΔHvap and the heat capacity (C), we can calculate the change in entropy (ΔS) using the equation:
ΔS = ΔHvap / T1 + C * ln(T2 / T1)
Step 2: Heating the vapor at constant pressure from 59.2 bar to 1 bar
During this step, the pressure remains constant, so there is no work done. Therefore, the change in entropy is only due to the heat added to raise the temperature.
The heat capacity (C) is given as 17.5 J/mol K, but we need to convert it to J/g K to match the units of ΔHvap.
Since we don't know the mass of the liquid, we can assume that we have 1 mole of the substance.
ΔS = C * ln(T2 / T1)
Step 3: Expanding the vapor from 1 bar to the final pressure
Similar to step 1, we can use the ideal gas equation to calculate the change in entropy during this step.
ΔS = R * ln(P2 / P1)
Now, sum up the entropy changes from each step to find the total change in entropy of the substance.
ΔS_total = ΔS1 + ΔS2 + ΔS3
That's how you can calculate the change in entropy for this specific problem. Please note that this explanation assumes that the substance behaves ideally and doesn't consider any other factors or non-ideal behavior that could affect the calculations.