Question

One atom of an element weighs \(3.32 \times 10^{-23} \mathrm{~g}\). How many number of gram atoms are there in \(20 \mathrm{~kg}\) of the element? (a) 2000 (b) 20 (c) 200 (d) 1000

Short answer

The correct answer is (c) 200.

Step-by-step solution

Understanding the concept of gram atoms

A gram atom is defined as the amount of an element whose mass in grams is numerically equal to its atomic mass. If one atom of an element has a mass of 3.32 x 10^-23 g, then one gram atom of that element would have a mass of the atomic mass of that element in grams. The atomic mass must first be determined to answer the problem.

Calculate the mass of one mole (one gram atom) of the element

Since the mass of a single atom is known, the mass of one mole (Avogadro's number, approximately 6.022 x 10^23 atoms) of atoms can be calculated. The formula to calculate the mass of one mole of the element is (mass of one atom) x (Avogadro's number).

Calculate the number of gram atoms in the given mass of the element

To determine the number of gram atoms in 20 kg of the element, divide the total mass of the element by the mass of one gram atom. This is done using the formula (Total mass of element in grams) / (Mass of one gram atom).

Key concepts

Avogadro's number

When we talk about Avogadro's number, we're diving into the fundamental aspects of chemistry. It's a constant that represents the number of entities, usually atoms or molecules, present in one mole of a substance. Now, let's make this clear: Avogadro's number is approximately equal to \(6.022 \times 10^{23}\).

This incredibly large number helps us relate the micro world of atoms to the macro world we interact with every day. Imagine you have a dozen eggs; that's \(12\) individual eggs. Similarly, if you have a mole of eggs (hypothetically speaking), you'd have Avogadro's number of eggs. But, instead of eggs, we are dealing with atoms. In the exercise, knowing Avogadro's number is crucial as it allows us to calculate the mass of one mole of the element from the mass of a single atom.

atomic mass

The concept of atomic mass is another cornerstone in chemistry. It is the weight of a single atom of an element, typically expressed in atomic mass units (amu). However, when we bridge the gap to practical measurements, we use grams. The atomic mass tells us how 'heavy' an atom is compared to a standard, with carbon-12 commonly being the reference point.

In simplified terms, if a single atom of an element weighs a certain amount, say \(3.32 \times 10^{-23}\) grams in our exercise, then the atomic mass of that element, in grams, will be the same number when multiplied by Avogadro's number. Understanding the atomic mass of an element is vital because it directly correlates to how many atoms of that element we'll end up with in a given mass – this ties directly to the exercise, as we are trying to determine how many 'gram atoms' comprise \(20\) kilograms of the element.

mole concept

Lastly, let's delve into the mole concept. This might conjure images of the small burrowing mammal, but in chemistry, a mole is a unit of measurement used to express amounts of a substance. It's similar in function to terms like 'pair' or 'dozen,' except it's used to count atoms, molecules, ions, or other elementary entities.

The mole concept is an extension of Avogadro's number and atomic mass. It enables chemists to count particles by weighing them. One mole of any substance will contain \(6.022 \times 10^{23}\) of its entities (Avogadro's number), and the mass of this mole is its atomic or molecular mass expressed in grams. The exercise showcases the mole concept by calculating how many 'gram atoms' or moles exist in \(20\) kg of an element, effectively translating the theoretical count of atoms into a practical, measurable mass.