Question

In a certain industrial process using a heterogeneous catalyst, the volume of the catalyst (in the shape of a sphere) is \(10.0 \mathrm{~cm}^{3}\). Calculate the surface area of the catalyst. If the sphere is broken down into eight spheres, each of which has a volume of \(1.25 \mathrm{~cm}^{3}\), what is the total surface area of the spheres? Which of the two geometric configurations of the catalyst is more effective? Explain. (The surface area of a sphere is \(4 \pi r^{2},\) in which \(r\) is the radius of the sphere.)

Short answer

The total surface area of the eight smaller spheres is greater than the surface area of the single bigger sphere. Therefore, breaking the sphere down into eight smaller spheres increases the effectiveness of the catalyst.

Step-by-step solution

Calculate the radius of the initial sphere

First, we need to convert the volume of the sphere into the radius. We can use the formula for the volume of a sphere, which is \(V = \frac{4}{3}\pi r^{3}\). Re-arranging for r, we find that \(r = \left(\frac{3V}{4\pi}\right)^\frac{1}{3}\). Substituting \(V = 10.0 \mathrm{~cm}^{3}\) into this formula, we find the radius of the initial sphere.

Calculate the surface area of the initial sphere

Next, we substitute the radius into the surface area formula: \(SA = 4 \pi r^{2}\), to find the surface area of the initial sphere.

Calculate the radius of the smaller spheres

Now we need to calculate the radius of the smaller spheres. We again use the formula for the volume of a sphere, substituting \(V = 1.25 \mathrm{~cm}^{3}\) into the formula, to find the radius of each smaller sphere.

Calculate the surface area of the smaller spheres

Substitute the found radius of the smaller spheres into the surface area formula to find the surface area of one smaller sphere. Then, multiply by 8 to find the total surface area of the eight smaller spheres.

Compare the surface areas in both configurations

Compare the total surface area of the single big sphere with the total surface area of the eight smaller spheres. The geometric configuration that has a larger surface area is more effective as a catalyst, since it offers more surface for the reactions to take place.

Key concepts

Heterogeneous Catalyst

Catalysts are substances that increase the rate of chemical reactions without being consumed in the process. A heterogeneous catalyst is one that exists in a different phase than the reactants — usually it is a solid that interacts with liquid or gas reactants. In the context of industrial processes, having a larger surface area can make a heterogeneous catalyst more effective. This is because more surface area allows for more contact between the reactants and the catalyst, facilitating more sites for the chemical reaction to occur.

Importance of Surface Area in Catalysis

When it comes to heterogeneous catalysis, the surface area plays a pivotal role. By breaking down a catalyst into smaller pieces or changing its shape to maximize surface area, the efficiency of the catalytic process can be significantly improved. In the exercise above, the effectiveness of the catalyst in different geometric configurations hinges on the amount of surface area available for reaction.

Volume of a Sphere

The volume of a sphere is a basic geometric concept used to determine the amount of space inside a spherical object. Mathematically, it is expressed using the formula \( V = \frac{4}{3}\pi r^3 \), where \( V \) stands for volume and \( r \) denotes the radius of the sphere. Understanding how to calculate the volume is crucial because it directly relates to the sphere's radius, which we can then use to determine the surface area — a key factor in the effectiveness of a spherical catalyst.

Application in Calculating Catalyst Size

To find the surface area of a spherical catalyst, it's necessary to first calculate its radius from the known volume. This computation is crucial for making comparisons between catalysts of different sizes and shapes because it provides us with the dimensions needed to further explore surface areas.

Geometric Configurations

The term geometric configurations relates to the shape and form of objects. In many scientific and engineering applications, the configuration of an object can drastically affect its performance. Specifically, for catalysts in chemical reactions, the geometric configuration can influence the reaction rate due to the variance in surface area.

Effect on Catalyst Performance

When comparing a single large sphere to several smaller spheres, even though they may have the same combined volume, the total surface area changes. Smaller spheres combined have a larger surface area than a single sphere of equivalent volume. In the exercise provided, breaking down the larger sphere into eight smaller ones increases the total surface area, potentially making the configuration of smaller spheres more effective for catalytic purposes, as they offer more area for reactants to interact with the catalyst.