Question

If Avogadro's number were to tend to infinity, the phenomenon of Brownian motion would (a) remain completely unaffected (b) become more vigorous than that observed with the present finite value of Avogadro's number, for all sizes of the Brownian particles (c) become more vigorous than that observed with the present finite value of Avogadro's number, only for relatively large Brownian particles (d) become practically unobservable, as the molecular impact would tend to balance one another for practically all sizes of Brownian particles

Short answer

As Avogadro's number tends towards infinity, Brownian motion would become practically unobservable for all sizes of Brownian particles.

Step-by-step solution

Understand Brownian Motion

Brownian motion is the erratic random movement of microscopic particles in a fluid, as a result of continuous bombardment from molecules of the surrounding medium.

Consider the Effect of Increasing Avogadro's Number

Avogadro's number (\( N_A \)) represents the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance — approximately (\(6.022 \times 10^{23}\) particles per mole). If Avogadro's number were to increase towards infinity, the number of molecular impacts on a Brownian particle would rise immensely.

Analyze the Effect on Brownian Motion

With an infinite number of molecular impacts from all directions, the impacts would tend to balance out due to the law of large numbers. This would result in a net force that is practically zero, making any observable Brownian motion unnoticeable.

Conclude the Effect on Brownian Motion

As Avogadro's number tends to infinity, the increased frequency of molecular collisions from all directions would lead to a balancing out of forces, resulting in an unobservable Brownian motion for all sizes of Brownian particles.

Key concepts

Avogadro's Number

Avogadro's number, denoted as \( N_A \), is a fundamental constant that is crucial in chemistry and physics. It is defined as the number of atoms in exactly 12 grams of carbon-12, which is approximately \( 6.022 \times 10^{23} \) atoms.

This large number helps chemists and physicists to count particles like atoms and molecules in a given sample by relating their mass to the number of particles. For instance, one mole of any substance always contains Avogadro's number of molecules or atoms, regardless of what substance you’re working with. This concept allows us to measure and work with the submicroscopic world in macroscopic quantities that can be easily weighed and handled.

Understanding Avogadro's number enhances the comprehension of molecular scale and quantity, which in turn informs other concepts, such as molecular impact, which is related to the behavior of particles like the ones seen in Brownian motion.

Molecular Impact

Molecular impact refers to the collisions exerted by molecules on a particle. These impacts are the driving force behind the phenomenon known as Brownian motion. In a fluid, countless molecules are in constant random motion due to thermal energy. When these molecules collide with larger particles, like pollen grains suspended in water, they impart a series of small pushes to the pollen grains.

While these impacts can be unequal, causing the pollen to move erratically, they might balance out over a long period of time or with a large enough quantity of impacts. The Brownian motion is, therefore, the observable result of countless molecular impacts that do not completely balance out immediately. If we were to consider a situation where Avogadro's number is infinitely large, the infinite number of molecular impacts on the Brownian particle would likely balance each other out, significantly dampening the motion that we observe.

Law of Large Numbers

The law of large numbers is a principle of probability according to which the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances. In the context of molecular impact and Brownian motion, as the number of impacts increases (with the hypothetical scenario of Avogadro's number tending to infinity), the overall effect of individual collisions will tend to balance out.

Imagine flipping a coin: the more flips you perform, the closer you will get to a 50/50 distribution of heads and tails. This is an illustration of the law of large numbers at work. Applying this to molecules striking a Brownian particle, if the number of impacts is exceedingly large - infinity, in our given exercise - it would result in an averaging effect where the particle feels uniform pressure from all directions, leading to a negligible net movement, and hence, an unobservable Brownian motion.