How to calculate partial molar volume from mole fraction of a mixture and its density ?
Answer:
The partial molar volume of a component is defined as the partial derivative of the total volume with respect to the amount of the component. It is equal to the partial derivative of the molar volume with respect to the molar fraction:
Vi = (∂V/∂ni) = ∂Vm/∂xi
The later definition is more useful for this problem
First convert the density data in to molar volume data by
Vm = M / ρ
M is the average molar mass of the mixture given by:
M = x₁·M₁ + x₂·M₂ = x₁·M₁ + (1-x₁)·M₂
Next step is to fit an appropriate equation to the data. Because x₂=1-x₁ its is sufficient to fit the molar volume as function of x₁. The type of the function depends on the data. I think in most cases a 2nd order polynomial is sufficiently precise. Whatever ever your equation is, it should match the pure component molar volumes at x₁=0 and x₁=1.
You will get the partial molar volumes of component 1 by calculating:
∂Vm/∂x₁ = V₁
The replace x₁ by (1-x₂) in the molar volume equation and calculate:
∂Vm/∂x₂ = V₂
An alternative would be a graphic procedure. Plot the data of the molar volume data versus x₁. You can find the partial molar volumina for a certain composition from the tangent to the plot at this composition.
Because Vm = x₁·V₁ + x₂·V₂ = x₁·V₁ + (1-x₁)·V₂
The equation of the tangent line at x₁= x₁' is
t(x₁) = x₁·V₁' + (1-x₁)·V₂'
Thus you can read the partial molar volumes V₁',V₂' at x₁' directly from the graph by:
t(x₁=0) = V₂' and
t(x₁=1) = V₁'
concentration=number of moles\volume in decimetre
Moles are measured by mass, not volume. Mole fraction is when a mole gets run over and end up as road pizza.
Hence the warning before you set out on a long road journey "Watch out for road pizza in the granny", which translates out as "Be careful of dead animal's in the slow lane."
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