Question

When concentrations of the reactants is increased sixteen times, the rate becomes two times. The reaction is of (a) \(1 / 4\) order (b) fourth-order (c) third-order (d) \(1 / 8\) order

Short answer

The reaction is of \(\frac{1}{4}\) order.

Step-by-step solution

Understand the Problem

To determine the order of the reaction, we need to relate how the rate of reaction changes with the concentration of the reactants. The question tells us that when the concentration is increased by 16 times, the rate increases by 2 times.

Recall the Reaction Rate Law

The rate of a reaction can be expressed as: \[ ext{Rate} = k[ ext{Reactant}]^n\]where \(k\) is the rate constant, \([ ext{Reactant}]\) is the concentration of the reactant, and \(n\) is the order of the reaction.

Set Up the Equation

Given that the concentration is increased 16 times and the rate is increased 2 times, we can set up the equation:\[2 = 16^n\]This equation arises from setting the new rate equal to the old rate times the increase factor from concentration, both in terms of power \(n\).

Solve for the Order

Solve the equation for \(n\):\[2 = 16^n\]Express 16 as a power of 2: \[16 = 2^4\]Substitute this in to get:\[2 = (2^4)^n = 2^{4n}\]Since the bases are the same, equate the exponents:\[1 = 4n \]Therefore, solve for \(n\):\[n = \frac{1}{4}\]

Verify the Answer

Substituting \(n = \frac{1}{4}\) into the expression confirms that the rate indeed becomes 2 times when the reactant is increased 16 times. Thus, the reaction is of \(\frac{1}{4}\) order.

Key concepts

Reaction Rate Law

A fundamental concept in understanding chemical reactions is the reaction rate law. This law helps us predict how the rate of a reaction changes as the concentration of reactants changes. Essentially, the rate of a reaction can be expressed by the equation:

• \( \text{Rate} = k[\text{Reactant}]^n \)

In this equation, \( k \) is the rate constant, a unique value to each reaction at a specific temperature, and \( [\text{Reactant}] \) represents the concentration of the reactant. The exponent \( n \) is called the order of the reaction. It is critical because it shows the dependency of the reaction's rate on the concentration of the reactant. By analyzing changes in the rate when concentrations change, one can deduce the order of the reaction.

Rate Constant

The rate constant, symbolized as \( k \), is a crucial component of the reaction rate law. It acts as a proportionality factor in the rate equation:

• \( \text{Rate} = k[\text{Reactant}]^n \)

The value of \( k \) does not change with the concentration of the reactant. However, it is affected by external factors such as temperature or the presence of a catalyst. Because of these dependencies, \( k \) is often used to understand the conditions under which a reaction occurs. Knowing \( k \) allows us to calculate the reaction rate for various reactant concentrations, provided the reaction order \( n \) is known.

Reactant Concentration

Reactant concentration is a key factor in the reaction rate law equation. It refers to how much of a reactant is present in a chemical reaction mixture at a given time. As seen in the formula:

• \( \text{Rate} = k[\text{Reactant}]^n \)

Changes in reactant concentration directly affect the rate of the reaction. In the provided exercise, the concentration of the reactant is increased sixteen times, causing the reaction rate to double. This change illustrates how varying reactant concentration can affect the outcome of the reaction based on the order \( n \). Thus, understanding reactant concentration is essential for predicting and controlling the speed of chemical reactions.

Exponent Solving

Being able to solve for the exponent in the reaction rate law is crucial for determining the reaction order. In our problem, one needs to solve the equation \( 2 = 16^n \) to find \( n \). To simplify this equation:

• First, express 16 as a power of 2: \( 16 = 2^4 \)
• Substitute into the equation: \( 2 = (2^4)^n = 2^{4n} \)
• Now that the bases are the same, equate the exponents: \( 1 = 4n \)
• Finally, resolve for \( n \): \( n = \frac{1}{4} \)

This calculation shows that the order of reaction is \( \frac{1}{4} \). Mastering exponent solving allows you to determine how the reactant concentration affects the reaction rate, a critical skill in chemical kinetics.