Question

Chemical Reactions: Increased temperature usually causes chemicals to react faster. If a certain reaction proceeds \(15 \%\) faster for each \(10^{\circ} \mathrm{C}\) increase in temperature, by what factor is the reaction speed increased when the temperature rises by \(50^{\circ} \mathrm{C} ?\)

Short answer

The reaction speed is increased by a factor of \(1.15^5\) when the temperature rises by \(50^{\text{C}}\).

Step-by-step solution

Understanding the problem

We need to determine the factor by which the reaction speed increases when the temperature is raised by \(50^{\text{C}}\). The reaction speed increases by \(15\text{%}\) for each \(10^{\text{C}}\) increase in temperature.

Convert percentage increase to decimal factor

First, convert the \(15\text{%}\) increase to a decimal by dividing by 100. This gives us \(0.15\). The factor for a \(10^{\text{C}}\) increase is \(1 + 0.15 = 1.15\).

Calculating the overall factor for a \(50^{\text{C}}\) increase

Since the temperature increases by \(50^{\text{C}}\), which is 5 intervals of \(10^{\text{C}}\), we need to multiply the factor of \(1.15\) by itself five times. This is equivalent to raising \(1.15\) to the 5th power, which is \(1.15^5\).

Compute the value of \(1.15^5\)

Use a calculator or another method to calculate the fifth power of \(1.15\). The result will give the factor by which the reaction speed is increased when the temperature is raised by \(50^{\text{C}}\).

Key concepts

Reaction Rate and Temperature

The rate at which chemical reactions occur is intrinsically linked to temperature, in most cases, a rise in temperature leads to an increase in reaction rate. This is described by the Arrhenius equation, which states that the reaction rate increases exponentially with an increase in temperature. The relationship between reaction rate and temperature is based on molecular kinetics; as temperature rises, molecules move faster, increasing the chances of collision and the energy with which they collide. More energetic collisions lead to a higher probability of reaction.

To quantify the increase in reaction rate with temperature, we usually express the temperature-rise factor in terms of a percentage. For example, if a reaction rate increases by 15% for each 10 degrees Celsius increase, this percentage must be converted into a factor that can be mathematically manipulated to determine the overall increase over a given temperature range, like a 50 degrees Celsius increase in our exercise.

Percentage to Decimal Conversion

Understanding percentage to decimal conversion is crucial in many aspects of science and mathematics, including calculating reaction rate increases in chemistry. To convert a percentage into a decimal, you simply divide by 100. This comes in handy when working with growth factors in equations because it allows for straightforward multiplication.

In the context of our exercise, the reaction rate increases by 15% for each 10°C. By converting 15% to a decimal, 15% divided by 100 equals 0.15. This decimal is then added to 1 to represent a 15% increase (since 100% equals 1 in decimal form), giving us a growth factor of 1.15 for each interval of temperature increase.

Exponential Growth in Chemistry

Exponential growth in chemistry refers to the process where an increase happens at a rate that is proportional to the current value—it accelerates over time. This is relevant in the context of reaction rates, where repeated application of a growth factor results in exponential increases. If the reaction rate increases by a factor for each unit of temperature increase, repeatedly applying that factor over several units of temperature increase compounds the effect.

To calculate the total increase over a 50°C increase, you raise the single-temperature increase factor (1.15) to the power of the number of 10-degree intervals in 50 degrees, which is five. This is mathematically represented as \(1.15^5\), demonstrating exponential growth, as the reaction rate is not simply added together, but rather multiplied in a compounded fashion to reflect consecutive percentage increases.