2 examples of molar volumes of binary mixtures

It‘s about the following task:

Figure: 2 examples of molar volumes of binary (2 components, i=A,B is the component index). mixtures. Let V be the total volume of the mixture. Using binary mixture as an example, explain/ define the molar volumes 𝑉m, 𝑣i, and 𝑉^E (excess). How can you read these quantities from the graph?

I really have no clue how to proceed here. I've been dealing with the task all day, but I still don't understand it. I have also watched countless videos about it, but none of them helped me. How can I read from the graph what Vm, vi and V^E are? I need a brief theoretical explanations of these things. I would be very glad if someone could help me further because I struggled the whole day with this task.

• Are you familiar with the concept of partial molar quantities? Commented Apr 3, 2023 at 16:53
• Do you know the equation for the molar volume of an ideal binary liquid solution in terms of the molar volumes of the pure components? Commented Apr 3, 2023 at 16:59
• And ideal solutions vs regular solutions? Commented Apr 3, 2023 at 17:12
• @ChetMiller, yes I'm familiar with it but I have no clue how to explain it theoretically. I only know the equation but I don't know the meaning of it. Everything was mathematically explained to us all the time, which is why I have difficulties to understand and solve this task. Commented Apr 3, 2023 at 20:07
• @Jon Custer, i only know that solutions are ideal when there is no interaction between the solvents. Commented Apr 3, 2023 at 20:19

2 Answers

For an ideal solution, the molar volume of the solution is a weighted average (in terms of mole fractions) of the molar volumes of the pure components at the same temperature and pressure as the mixture: $$V^{ID}=V^0_Ax_A+V^0_Bx_B=V^0_A+(V^0_B-V^0_A)x_B$$The excess molar volume is equal to the actual molar volume minus the ideal molar volume: $$V^{EX}=V_m-V^{ID}$$or$$V_m=V^{ID}+V^{EX}$$

• Alright, thank you for the equations but how should I read these quantities from the graph? I have difficulties to understand the meaning of these equations. What do these quantities explain? Where on the graph are they? How can I read these quantities from the graph and why? Commented Apr 4, 2023 at 3:26
• The questions you are asking involve Middle-School math (elementary algebra), not Physics. Didn't. they teach this to you in Middle School, how to plot a straight line and get the intercepts? Commented Apr 4, 2023 at 8:42
• The straight dashed lines are for the ideal solution behavior. Commented Apr 4, 2023 at 8:52
• I found it after I posted my question. I solved it now. Commented Apr 4, 2023 at 9:33
• But there are other questions like: Are there areas with negative partial volumes, if so, where (which of the mapped mixtures, which component, which areas)? Is this possible at all with the volume of mixtures? How should I proceed here? Schould I draw a tangent line? Commented Apr 4, 2023 at 10:06

This is how to get the equation for the tangent line. The starting equations are $$[V_m]_{x=x_0}=(1-x_0)[V_A]_{x=x_0}+x_0[V_B]_{x=x_0}$$and$$\left[\frac{dV_m}{dx}\right]_{x=x_0}=[V_B]_{x=x_0}-[V_A]_{x=x_0}$$where $$x=x_0$$ is the point of tangency and $$[V_A]_{x=x_0}$$ and $$[V_B]_{x=x_0}$$ are the partial molar volumes at $$x=x_0$$. So, using the point-slope formula for the equation of a straight line, we have the equation of the tangent line as $$V=[V_m]_{x=x_0}+\left[\frac{dV_m}{dx}\right]_{x=x_0} (x-x_0)$$

Combining the previous equations, the equation for the tangent line is $$V=(1-x)[V_A]_{x=x_0}+x[V_B]_{x=x_0}$$This indicates that the partial molar volumes of A and B at the specified tangent point are equal to the intercepts of the tangent line at x = 0 and x = 1.

• Thanks a lot! But I have further questions. Is there a negative VL in any of the mixtures? If so, in which areas? Can this be at all? Commented Apr 9, 2023 at 14:35
• For that I need to draw a tangent line, but how do I know if it‘s negative? How should I show this graphically? Commented Apr 9, 2023 at 14:35
• The intercepts of the tangent line are the partial molar volumes. Commented Apr 9, 2023 at 14:48
• Yes, but how do I figure out if they are graphically negative? Commented Apr 9, 2023 at 15:05
• I mean in which areas on the graph is V_l (the partial molar volume) negative ? Commented Apr 9, 2023 at 15:13