Question

On a PV diagram, suppose we place volume on the `y` axis; is the slope negative or positive?

Answer 1

Negative... and nonconstant.

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Well, you can start from Boyle's law:

`P_1V_1 = P_2V_2`

at constant temperature and mols of gas. Hence,

`V_2 = P_1/P_2 V_1`

If `P_2 > P_1`, then we see that the volume decreases from left to right, as it should. That clearly eliminates `c` and `d`.

The slope would be given by:

`"slope" = (DeltaV)/(DeltaP) -= (V_2 - V_1)/(P_2 - P_1)`

Now we just need to show that this slope is NOT a constant. Try subtracting `P_1V_2` from Boyle's law:

`P_1V_1 - P_1V_2 = P_2V_2 - P_1V_2`

`=> P_1(V_1 - V_2) = V_2(P_2 - P_1)`

`=> -P_1DeltaV = V_2DeltaP`

`=> color(blue)(barul(|stackrel(" ")(" "(DeltaV)/(DeltaP) = -V_2/P_1" ")|))`

The slope is negative, which makes sense. If pressure increases at constant temperature and mols of gas, the volume should compress.

Since `P_1` does not ever change, but `V_2` will decrease as `P_2` increases, we can see that the slope becomes less and less negative, and is surely NOT constant. In fact, it starts at a high negative and becomes a small negative, not unlike a `1//x` curve in quadrant I.

Clearly, it means the slope decreases over time and is negative.