In gas law equations, if volume and temperature are directly proportional, why can't we write the equation as temperature divided by volume? [closed]








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In gas law equations, volume, temperature, pressure and amount of substance may vary. In the simpler equations, like Charles' and Amonton's, temperature and pressure/volume are directly proportional. Why are the equations specifically with the temperature in the denominator, like this one with the reciprocal being untrue ( I'm asking a conceptual, not mathematical question)?


$$\frac{V\_1}{T\_1} = \frac{V\_2}{T\_2}$$




Interesting! This bothered me long time ago in elementary school. Anyway, later I learned elsewhere that the mathematical convention is to write the dependent variable first in ratio and proportion with a proportionality symbol. Such relations appeared all the time in chemistry and physics. It is very sad that modern high-school algebra has eliminated entire chapter on ratio and proportion properties (esp. see *invertendo* property\*).


So, $$ V \propto T $$ which means


$V$ is a function of $T$, not the other way round. Temperature dictates what the volume of gas will be.


$$ V =c T $$


where $c$ is a constant of proportionality. Following the graphing convention, this means that one will plot $T$ on the x-axis and volume $V$ on the y-axis. This is nothing but a linear equation with zero intercept when $T$ is in Kelvins.


We can also write,


$$\frac{ V\_1}{T\_1} =c $$


and
$$\frac{ V\_2}{T\_2} =c $$


Since both fractions are equal to $c$,


$$\frac{ V\_1}{T\_1} =\frac{ V\_2}{T\_2} $$


\****Invertendo*** property of ratios tells us that a/b: c/d is the same as b/a:d/c.


$$\frac{ T\_1}{V\_1} =\frac{ T\_2}{V\_2} $$


is also correct. It is not wrong algebraically. Here you don't care which one is the independent variable.


As another unrelated example is Ohm's law:


One can say, $$ I \propto Voltage $$ which means current depends on the applied voltage.


So, $$ I = c Voltage $$


$c$ turns out to be the inverse of resistance $R$.


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The questions in the title and the body ask different things.


We cound write $\frac{T}{V}$ instead of $\frac{V}{T}$, but there is, perhaps unwritten, convention using the cause as the denominator.


The equation $\frac{V\_1}{T\_1}=\frac{V\_2}{T\_2}(=\frac{nR}{p})$ is obviously valid only for ideal gases ($pV=nRT$) at isobaric conditions.


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