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

##### 1 Answer

Negative... and nonconstant.

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

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_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 *decrease* as *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

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