Max inversion temperature for a gas (Dieterici’s equation of state)

[jonny997]

Homework Statement

I’m trying to calculate the maximum inversion temperature from the inversion curve.

Relevant Equations

See below

The notes my lecturer has provided state that the maximum temperature can be found taking p = 0 in the inversion curve formula, given as:

I’m not sure how to obtain this??

These are the formulas:

This is my attempt at a solution :

Not sure if this approach is right?

 

[TSny]

Your work looks correct to me.

For part (b) you obtain the inversion curve as a relation between the inversion temperature $T$ and the molar volume $V$:

I believe an inversion curve is typically a relation between $T$ and $P$ rather than between $T$ and $V$.

 

[jonny997]

Your work looks correct to me.

For part (b) you obtain the inversion curve as a relation between the inversion temperature $T$ and the molar volume $V$:

View attachment 343722

I believe an inversion curve is typically a relation between $T$ and $P$ rather than between $T$ and $V$.

Hey. Thanks for the response. Do you have any idea how I would go about obtaining the expression my lecturer has provided? Namely, $$ P_{inv} = \left[\frac{2a}{b^2} - \frac{RT}{b}\right]e^{\frac{1}{2}-\frac{a}{RTb}}$$ If I am to rewrite $T = \frac{2a}{R}\left(\frac{1-\frac{b}{V}}{b}\right)$ in terms of $P$ do I not need an explicit expression for T or V from Dieterici’s e.o.s.

 

[TSny]

If I am to rewrite $T = \frac{2a}{R}\left(\frac{1-\frac{b}{V}}{b}\right)$ in terms of $P$ do I not need an explicit expression for T or V from Dieterici’s e.o.s.

You need to eliminate $V$ between the equation of state and the equation $T = \frac{2a}{R}\left(\frac{1-\frac{b}{V}}{b}\right)$. Decide which equation is easier to solve for $V$ and then substitute for $V$ in the other equation.

 

[jonny997]

You need to eliminate $V$ between the equation of state and the equation $T = \frac{2a}{R}\left(\frac{1-\frac{b}{V}}{b}\right)$. Decide which equation is easier to solve for $V$ and then substitute for $V$ in the other equation.

Ahhh okay, thank you. For some reason it didn’t click that I could solve for V using $T = \frac{2a}{R}\left(\frac{1-\frac{b}{V}}{b}\right)$ and then sub that into the other equation I’ve got it now