Which of the following is consistent with Avogadro’s law?

Keep in mind that Avogadro's Law is used to describe the behavior of an ideal gas under a very specific set of circumstances.

More specifically, in order for Avogadro's Law to apply, the temperature and the pressure of the gas must be kept constant.

Avogadro's Law states that when the temperature and the pressure of a gas are kept constant, the volume of the gas is directly proportional to the number of moles present in the sample.

#color(blue)(ul(color(black)("when T and P are constant" => V prop n)))#

In other words, when the number of moles of gas increases, the volume of the gas will increase as well. Similarly, when the number of moles of gas decreases, the volume of the gas will decrease as well.

So right from the start, options A) and B) are eliminated. Now, the direct relationship that exists between #V# and #n# when #T# and #P# are constant can be expressed as

#color(blue)(ul(color(black)(V/n = "constant")))#

Why is that the case?

Notice that when #V# increases, the only way for the ratio #V/n# to be constant is if #n# increases by the same factor as #V#.

This is not what happens for

#V * n = "constant"#

In this case, when #V# increases, the only way for the product #V * n# to be constant is if #n# decreases by the same fact as #V#.

Therefore, you can say that option D) is correct.

You can get the same result by rearranging the ideal gas law equation

#color(blue)(ul(color(black)(PV = nRT)))#

to isolate the constants on one side of the equation. In this case, you will have

#PV = nRT#

#V = (nRT)/P#

Divide both sides by #n# to get

#V/n = (RT)/P implies color(blue)(ul(color(black)(V/n = "constant")))#

So remember, Avogadro's Law applies only to cases where the pressure and the temperature are constant.