Question

If \(m_{A}\) gram of a metal \(A\) displaces \(m_{B}\) gram of another metal \(B\) from its salt solution and if the equivalent weights are \(E_{A}\) and \(E_{B}\) respectively then equivalent weight of \(A\) can be expressed \(\begin{aligned}&\text { as: } \\\&\text { (a) } E_{A}=\frac{m_{A}}{m_{B}} \times E_{B} \quad \text { (b) } E_{A}=\frac{m_{A} \times m_{B}}{E_{B}} \quad \text { (c) } E_{A}=\frac{m_{B}}{m_{A}} \times E_{B} & \text { (d) } E_{A}=\sqrt{\frac{m_{A}}{m_{B}} \times E_{B}}\end{aligned}\)

Short answer

The correct expression for the equivalent weight of metal A is (a) \( E_{A} = \frac{m_{A}}{m_{B}} \times E_{B} \).

Step-by-step solution

Understanding the Displacement Reaction

In a displacement reaction, one metal displaces another metal from its solution. The amount of metal displaced is directly proportional to the equivalent weight of the metal that caused the displacement. This relation is based on the law of chemical equivalence.

Law of Chemical Equivalence

According to the law of equivalence, the weight of a substance (metal in this case) is directly proportional to its equivalent weight. Mathematically, we can express this in terms of the displaced metal (B) and the displacing metal (A): \( m_{A} / E_{A} = m_{B} / E_{B} \), where \( m_{A} \) and \( m_{B} \) are the masses and \( E_{A} \) and \( E_{B} \) are the equivalent weights of metal A and B, respectively.

Calculating the Equivalent Weight of Metal A

From the previous step, by cross-multiplication, we get: \( m_{A} / E_{A} = m_{B} / E_{B} \) or, \( E_{A} = (m_{A} / m_{B}) \times E_{B} \). This formula gives us the equivalent weight of metal A in terms of the mass of both metals and the equivalent weight of metal B.

Key concepts

Displacement Reactions

Understanding displacement reactions in chemistry is vital because they underpin a vast array of reactions involving metals and their compounds. Displacement reactions occur when a more reactive metal can replace a less reactive metal in a compound, typically a salt solution. In these reactions, the more reactive metal takes the place of the less reactive one, demonstrating a fundamental property of metals - their reactivity series.

For example, when zinc metal is added to a copper sulfate solution, a displacement reaction occurs. The zinc displaces the copper from the sulfate, resulting in zinc sulfate and copper metal. The reactiveness of the metals determines the direction of the displacement, adhering to the predictable order outlined in the reactivity series.

Understanding these reactions is crucial because they not only occur in laboratories but also in industrial processes and nature, such as the corrosion of metals.

Law of Chemical Equivalence

The law of chemical equivalence is a cornerstone concept in quantitative chemistry, especially when analyzing displacement reactions. It posits that substances react in proportion to their equivalent weights, providing a quantitative relationship between reactants and products. The equivalent weight of a substance is its mass that will combine with or displace a fixed quantity (usually 1 mole) of another substance.

To exemplify, the chemical equation \( A + B \rightarrow AB \) follows this law. If \( A \) and \( B \) have equivalent weights of \( E_A \) and \( E_B \) respectively, when \( m_A \) grams of \( A \) reacts with \( m_B \) grams of \( B \) their equivalent weight ratio will be maintained as \( \frac{m_A}{E_A} = \frac{m_B}{E_B} \). This law is foundational when performing stoichiometric calculations in chemical reactions.

Chemical Stoichiometry

Chemical stoichiometry is key to understanding the proportions in which chemicals react. This discipline within chemistry focuses on the quantitative relationship between substrates and products in a chemical reaction, using the mole concept as a cornerstone for its calculations. Stoichiometry relies on balanced chemical equations to dictate the amount of reactants necessary to produce a desired quantity of product.

By applying stoichiometry, chemists can predict the outcome of a given reaction, determine limiting reactants, and calculate yields — essential tasks in fields from pharmaceutical synthesis to environmental monitoring. For instance, if you need to produce a specific amount of a product, stoichiometry provides the exact quantities of reactants needed, ensuring efficiency and cost-effectiveness in chemical production.

Equivalent Weight Calculation

Equivalent weight calculation often confuses students, yet it's crucial for understanding various aspects of chemistry, including titration and the stoichiometry of reactions. Equivalent weight represents the mass of a substance that reacts with or is equivalent to eight grams of oxygen, one gram of hydrogen, or one mole of electrons in redox reactions.

To calculate the equivalent weight of a metal participating in a displacement reaction, the mass of the displacing metal (\( m_A \) grams), the mass of the displaced metal (\( m_B \) grams), and the equivalent weight of the displaced metal (\( E_B \)) are used. The equation \( E_A = \frac{m_A}{m_B} \times E_B \) depicts this calculation accurately. By understanding and applying this formula, students and chemists can engage with a host of practical applications, from analyzing water hardness to processing industrial waste.