Problem 23 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A mixture of \(2 \times 10^{21}\) molecules of \(\mathrm{P}\) and \(3 \times 10^{21}\) molecules of \(\mathrm{Q}\) weighs \(0.60 \mathrm{~g}\). If the molecular mass of \(\mathrm{P}\) is 45, the molecular mass of \(\mathrm{Q}\) will be \(\left(N_{\mathrm{A}}=6 \times 10^{23}\right)\) (a) 45 (b) 180 (c) 90 (d) 270

Step-by-Step Solution

Answer

The molecular mass of Q is 90 g/mol.

Step 1: Calculate total number of moles of P

Use Avogadro's number to convert the number of molecules to moles for substance P. The formula is: Number of moles (n) = Number of molecules / Avogadro's number. For P: \( n_P = \frac{2 \times 10^{21}}{6 \times 10^{23}} \) moles.

Step 2: Calculate the mass of P

Multiply the number of moles of P by its molecular mass to find the mass of P. \( m_P = n_P \times \text{Molecular mass of P} \) which is \( m_P = \frac{2 \times 10^{21}}{6 \times 10^{23}} \times 45 \) g.

Step 3: Calculate the mass of Q

Subtract the mass of P from the total mass of the mixture to find the mass of Q. \( m_Q = \text{Total mass} - m_P \).

Step 4: Calculate total number of moles of Q

Use Avogadro's number to convert the number of molecules to moles for substance Q. \( n_Q = \frac{3 \times 10^{21}}{6 \times 10^{23}} \) moles.

Step 5: Calculate the molecular mass of Q

Divide the mass of Q by the number of moles of Q to find its molecular mass. \( \text{Molecular mass of Q} = \frac{m_Q}{n_Q} \).

Step 6: Determine the correct answer

Compute values from previous steps and match the result with the given options to find the molecular mass of Q.

Key Concepts

Avogadro's Number

Understanding Avogadro's number is fundamental in chemistry, especially when dealing with molecular mass calculations. Avogadro's number, denoted as \( N_A \), is the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. The value of Avogadro's number is approximately \( 6.022 \times 10^{23} \) particles per mole.

When you're given a number of molecules and you need to find out how many moles they represent, you divide the number of molecules by Avogadro's number. It’s like converting dozens into eggs – if you know that one dozen equals 12 eggs, to find out how many dozens 24 eggs represent, you simply divide 24 by 12. Similarly, to convert molecules into moles, you divide by Avogadro’s number. This conversion is crucial for solving many chemistry problems.

Moles Conversion

The mole is a central unit in chemistry that allows chemists to count atoms, molecules, and other particles in a given sample. To convert molecules to moles, you need to employ the relationship given by Avogadro's number. The formula for this conversion is:
\[\text{Number of moles} = \frac{\text{Number of molecules}}{\text{Avogadro's number}}\]
A thorough understanding of this process is essential for mole calculations and relates directly to the concepts of molar mass and mass conversions.

For instance, in our exercise, we calculate the moles of substance P by dividing the number of molecules by Avogadro's number. Once you have the moles, you can determine the mass of a substance by multiplying the moles by the molecular mass (the mass of one mole of a substance). This two-step process is fundamental in solving chemistry problems involving mass and molecular mass calculations.

Chemistry Problems

Addressing chemistry problems often involves a systematic approach to break down complex concepts into manageable steps. For instance, when calculating the molecular mass of a substance, the procedure typically requires determining the number of moles and the mass of a substance first, as seen in the exercise provided.

This particular problem showcases how to use given data - the number of molecules of two substances and the total mass of a mixture - to find the molecular mass of an unknown compound. It is a common type of problem in chemistry that tests your understanding of molecular mass calculation, moles conversion, and the use of Avogadro's number.

A critical tip for students: always ensure your units are consistent and double-check your calculations at each step. Such vigilance helps avoid simple errors and ensures accurate results. By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of chemistry problems.