Is the molarity of a pure substance always lower than the corresponding molality?

Let's take water, for example.

STP is #25^oC# and #1 "bar"#. In these conditions, the density of water is #0.9970479 g/(cm^3)#.

Let's take pure water and find its molarity concentration. #1.000 L# water would be:

#[(1.000 cancel(L)*(0.9970479 cancel(kg))/(1cancel(L)))((1000cancel(g))/(1 cancel(kg))) ((1 mol)/(18.015 cancel(g)))]/(1.000 L) = 55.345 M#

We don't usually say this concentration because it's so high.

Now let's find its molality concentration. #1.000 L# water would be:

#[(1.000 cancel(L)*(0.9970479 cancel(kg))/(1cancel(L)))((1000cancel(g))/(1 cancel(kg))) ((1 mol)/(18.015 cancel(g)))]/(0.9970479 kg) = 55.509 m#

In comparing molarity and molality, if we made the molality the same value as the molarity (#55.345 m# and #55.345 M#), we would be decreasing the molality and increasing the #kg# of water used ("dilution"), so now we theoretically have a greater volume of water to achieve the same molality as a specific molarity value.

So, in the case of pure water, its molarity has the lower volume. However, it really depends on the density of the substance. If its density is less than 1, then the same molarity has a lower volume than the same molality.