Question

The atomic masses of two elements \(\mathrm{P}\) and Q are 20 and 40 , respectively. If ' \(a\) ' \(\mathrm{g}\) of \(\mathrm{P}\) contains ' \(b\) ' atoms, how many atoms are present in ' \(2 a\) ' \(\mathrm{g}\) of \(\mathrm{Q}\) ? (a) \(a\) (b) \(b\) (c) \(2 a\) (d) \(2 b\)

Short answer

The number of atoms present in '2a' grams of Q is 'b'.

Step-by-step solution

Understanding the Molar Mass Relationship

The first step is to understand that the number of atoms in a sample is directly proportional to its mass, divided by its molar mass (atomic mass expressed in grams). For element P, the molar mass is 20 g/mol. The mass of P given is 'a' grams, which contains 'b' atoms.

Calculating the Moles of P

Calculate the number of moles of P represented by 'a' grams. Number of moles of P = mass of P / molar mass of P = a / 20.

Finding the Proportion of Atoms in P

If 'a' grams of P has 'b' atoms, then by proportion, 1 gram of P has b/20 atoms. Therefore, 20 grams of P, which is 1 mole of P, has 20 * (b/20) = b atoms. This confirms that 1 mole of any element has Avogadro's number of atoms.

Applying the Same Logic to Q

For Q, the molar mass is 40 g/mol. The mass of Q given is '2a' grams. The number of moles of Q in '2a' grams is (2a) / 40 = a / 20, which is the same number of moles as 'a' grams of P.

Relating Moles of Q to Atoms

Since 1 mole of P contains b atoms and the number of moles in '2a' grams of Q is the same as 'a' grams of P, '2a' grams of Q also contains b atoms.

Key concepts

Atomic Mass

The atomic mass of an element is a fundamental property that reflects its mass on an atomic scale. It is commonly expressed in atomic mass units (amu), with one amu roughly equivalent to the mass of a proton or neutron. The importance of atomic mass in chemistry cannot be overstated, as it is used to determine the mass of atoms in a given element.

Understanding atomic mass is crucial for a variety of calculations in chemistry, including determining the number of atoms in a sample. Let's consider an example. The atomic mass of an element, P, is 20, which means each atom of P has a mass of 20 amu. If we scale this up from individual atoms to moles, which is where Avogadro's number comes into play, we can bridge the gap between the microscopic world of atoms and the macroscopic world we can measure.

Avogadro's Number

Avogadro's number, approximately \(6.022 \times 10^{23}\), is one of the cornerstones of chemistry. This number represents the quantity of particles, usually atoms or molecules, in one mole of a substance. Named after Amedeo Avogadro, this constant allows chemists to count particles by weighing them. It is analogous to a dozen eggs being a countable representation of the concept 'dozen', Avogadro's number is a much larger 'dozen' for atoms and molecules.

Every mole of any element is understood to contain Avogadro's number of atoms. Therefore, when working with atomic masses and amounts in grams, we relate the macroscopic measurements to the atomic scale by converting them to moles, which are directly connected to Avogadro's number. By using this constant, we can calculate the number of atoms in any given mass of an element. For example, a mole of element P with atomic mass 20 would contain \(6.022 \times 10^{23}\) atoms.

Molar Mass Calculation

Molar mass calculation is a practical tool for converting between the mass of a substance and the amount in moles. Molar mass is defined as the mass of one mole of a substance, generally measured in grams per mole (g/mol). For instance, if element P has an atomic mass of 20 amu, its molar mass is 20 g/mol.

To calculate the number of moles from a given mass, one uses the formula:

\[\text{Number of moles} = \frac{\text{given mass (g)}}{\text{molar mass (g/mol)}}\].
This formula allows us to determine the number of moles in 'a' grams of element P. By converting mass to moles, and recognizing that each mole contains Avogadro's number of atoms, we can also determine the number of atoms in the given mass.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It relies on the law of conservation of mass, which states that matter is neither created nor destroyed in a chemical reaction. Stoichiometry involves calculations of mass and moles which allow chemists to predict the amounts of substances needed or produced in a reaction.

In the context of our example comparing elements P and Q, we would use stoichiometry to relate the masses and atomic counts of each element. By understanding that moles are the bridge between the mass of a substance and the number of constituent atoms or molecules, stoichiometry allows us to solve for unknown quantities, such as the number of atoms in '2a' grams of Q. It's a discipline that meticulously quantifies the dance of atoms during chemical transformations.