Graphing x/y against y

Question

This is a question about graphing the relationship between two variables. It is about the ideal gas equation $$pV=nRT$$ to plot ${p}\over{T}$ against $T$, where ${nR}\over{V}$ is constant. How would this graph look like?

And in general, how do you plot ${x}\over{y}$ against $y$?

if you're asking how to plot $x/y=y$ then just multiply both sides by $y$ and you have yourself a parabola. — Jemmy, Nov 21 '14 at 15:33
Actually, this is a question related to the ideal gas equation ($pV = nRT$). I need to plot ${p}\over{T}$ against $T$, where ${nR}\over{V}$ is constant. How would this graph look like? — Plastic Astronaut, Nov 21 '14 at 15:36
well first off if $y^2=x$ then $y=\pm\sqrt{x}$, not just $\sqrt{x}$ which is indeed a parabola, just flipped on a different axis. As for your equation, I'm not sure what you mean. Is $p$ held constant? — Jemmy, Nov 21 '14 at 15:41
@Jeremy p is not held constant, only n, R and V are held constant. — Plastic Astronaut, Nov 21 '14 at 15:42
Before I post my answer, Are you absolutely sure that that $\frac{1}{V}$ is constant? — Nick, Nov 22 '14 at 02:43
Dear @Jeremy, to plot $f(x)$ against $g(x)$ means that we must find all points $\Big(g(x),f(x)\Big)\in\Bbb R^2$. If you're unfamiliar with this concept: Check out Indicator Diagrams — Nick, Nov 22 '14 at 09:02

score 0 · Answer 1 · answered Nov 21 '14 at 16:02

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I don't know why you want to do that, but plotting $p/T$ vs $T$, you put $T$ on the horizontal axis and $p/T$ on the vertical axis and plot. You get a hyperbola, just like $y=1/x$ In your ideal gas case, if $\frac {nR}V$ is constant, so $p/T$ is, too.

answered Nov 21 '14 at 16:02

Ross Millikan

374,822

Why would it be a hyperbola? – Plastic Astronaut Nov 21 '14 at 16:05
Because when you say "plot against T", you are putting T on the horizontal axis. The value of the function is p/T, which goes on the vertical axis. There is no magic to the names of the axes-we just typically use $x$ horizontal and $y$ vertical. Look at this Alpha plot – Ross Millikan Nov 21 '14 at 16:23
Yes, but isn't p/T varying as T is? – Plastic Astronaut Nov 21 '14 at 16:30
Of course. That is why you are plotting a graph. You plug values of $T$ in to get points $(T,p/T)$ and plot them. – Ross Millikan Nov 21 '14 at 16:40
In the given isochoric setting, $P \propto T$. Hence, $P$ is not constant but $\frac{P}{T}$ is. Infact, $P$ can be considered a function of $T$ as $P(T)$. My point is, the function $f(T) = \frac{P(T)}{T}$ won't get you a hyperbola but instead it will be a constant function similar to say $f(x) = \pi$. – Nick Nov 22 '14 at 08:56

score 0 · Answer 2 · answered Nov 22 '14 at 02:30

The ideal gas law tells us that p/T is equal to nR/V, always, for an ideal gas. When using this law, it may helpful to keep in mind that all of the quantities are positive.

When nR/V is constant, so is p/T, because they are always equal. The plot of p/T vs T will be a horizontal straight line. On the usual temperature scales, the temperatures are only positive, so the horizontal straight line should not cross the vertical axis of the plot.

In general, we might have a function y = f(x). If we wanted to plot x/y vs y, we would need to know the function f(x).

Graphing x/y against y

2 Answers2