Interpreting Data to Satisfy Conservation of Mass

[swmmr1928]

Homework Statement

This is not homework. I am having trouble interpreting this data. I don't see how it satisfies a conservation of mass. As the temperature is increasing, how is acetone increasing its mole fraction in both phases? If the mole fraction of acetone increases in one phase, then shouldn't it decrease in the other phase? Mole fraction is a conserved quantity right?

Homework Equations

Perry Handbook

The Attempt at a Solution

see [1]

 

[TSny]

Hi swmmr1928. I'm fairly sure "mole fraction of A in liquid phase" means the ratio n_A^l/(n_A^l+n_C^l) where n refers to number of moles and the superscript "l" refers to liquid phase. "A" is for acetone and "C" is for chloroform.

Likewise, "mole fraction of A in vapor state" meansn n_A^v/(n_A^v+n_C^v) where "v" refers to vapor state.

With this definition, you can see that the sum of the mole fractions of A in liquid and vapor phases does not need to remain constant as temperature changes.

Here's a little exercise. (Numbers are totally made up!) Consider a system of 10 moles of A and 10 moles of C initially with both substances entirely liquid. Slowly raise the temperature until some of the liquid begins to vaporize. Suppose at temperature T₁ we find

n_A^l = 9.9 moles n_A^v = 0.1 moles (Note total is still 10 moles)
n_C^l = 9.98 moles n_C^v = 0.02 moles

Note that the vapor is richer in acetone because acetone is more volatile than chloroform.

Now we raise the temperature to T₂ and find

n_A^l = 6.0 moles n_A^v = 4.0 moles
n_C^l = 8.5 moles n_C^v = 1.5 moles

Calculate the mole fractions of A in the liquid and vapor phases for these two temperature. Do you find that the mole fractions of A in both phases decrease with increase in temperature?

 

[Chestermiller]

An easy way to get a feel for this is to consider two liquids that form an ideal solution when mixed, and whose vapors form an ideal gas solution in the gas phase. Your two substances do not quite do this, but that won't detract from the analysis for the ideal case.

Let x_a and x_b represent the mole fractions of the two species in the liquid phase, and let y_a and y_B represent the mole fractions of the two species in the gas phase. Let P represent the total pressure, which in your situation is a constant. Let p_a (T) be the equilibrium vapor pressure of pure a at temperature T, and p_b (T) be the equilibrium vapor pressure of pure b at temperature T. Then for the ideal situation above,

P y_a = p_a x_a
P y_b = p_b x_b

If you add these two equations together, you get:

P = p_a x_a + p_b (1 - x_a)

Solving for x_a, you then get:

x_a= (P - p_b) / (p_a - p_b)

For a given temperature, this equation gives the mole fraction of a in the liquid phase. The mole fraction of a in the gas phase is then given by:

y_a = p_a x_a / P

In many cases, the ratio of the equilibrium vapor pressures p_a / p_b is fairly insensitive to temperature (i.e., assuming nearly matching heats of vaporization of the pure substances). This is called the assumption of constant "relative volatility." If we make this approximation, and apply it to the first two equations, we obtain:

y_a/(1 - y_a) = x_a/ (1 - x_a)(p_a / p_b)

According to this equation, y_a would increase monotonically with x_a under the constant relative volatility approximation.