Mmmmmm Something very interesting I found about inequalities

[drtg45]

So I was brainstorming trying to find what I did wrong in this chemistry problem:

https://www.physicsforums.com/showthread.php?t=627731 (simple algebra no chemistry knowledge required)

And I noticed something peculiar about the process of solving inequalities, let me sum up in a phrase what I found:

"When solving inequalities the VERY FIRST thing we should do is input the known values of all the variables we have in the inequality"

In the problem in the previous link what I did was solve for T first and then at the very end input all the values to get a numerical answer (just as I have been doing all my life with equations), doing it that way I got a wrong answer because there was a change of direction in the inequality sign:

ΔG=ΔH-TΔS

0>ΔH-TΔS

-ΔH>-TΔS

\frac{-ΔH}{-ΔS}<T (the inequality sign changes direction when we multiply or divide both sides by a negative number right?)

\frac{ΔH}{ΔS}<T (signs cancel each other)

\frac{-114.1kJ}{-146.4·10^-3kJ/K}<T (we input the values)

780K<T

-----------------------------------------------------------------------------------------

In contrast when I repeat the problem inputting the values first I get the correct answer:

ΔG=ΔH-TΔS

0>ΔH-TΔS

0>-114.1kJ-T(-146.4·10^{-3}kJ/K)

114.1kJ>-T(-146.4·10^{-3}kJ/K)

114.1kJ>T(146.4·10^{-3}kJ/K)

\frac{114.1kJ}{146.4·10^-3kJ/K}>T

780K>T , indeed when the temperature is lower than 780K the inequality 0>ΔG is satisfied, ΔG is negative, and the reaction is spontaneous!

So please someone aware me on this, is the fact that the very first step in solving an inequality is to input the values and then continue with the solving process kind of an unwritten rule or something? Because I never heard of it! then again I don't think I ever solved inequalities with more than one variable.

 

[jgens]

Notice that your ΔS is negative, so when you divide by -ΔS you are actually dividing by a positive number, and therefore the ">" sign does not change. The issue is less with inputting values first as it is with paying close attention to your signs.

 

[acabus]

It's certainly easy, from a practical point of view, to put in the values first, so you know if you're multiplying/dividing by a negative number, which would change the inequality sign. Mathematically, if you consider separate cases for each variable being positive or negative, it makes no difference.

 

[drtg45]

Notice that your ΔS is negative, so when you divide by -ΔS you are actually dividing by a positive number, and therefore the ">" sign does not change. The issue is less with inputting values first as it is with paying close attention to your signs.

That would require foreshadowing of the values of the variables, which means constantly looking back and forth at the known data which in my opinion can be distracting (at least for me) when your mind is working on a different qualitative realm (imagining atoms bouncing around, atomic bonds being broken and formed, which atoms bonds with which and why, the vibrational frequency of the molecules, etc...) as opposite to the quantitative computational math realm. Maybe it is better to just input the values in the beginning and get the foreshadowing and looking back and forth over with!