Problem 32 - Chemistry The Science in Context

Question

Pieces of different metals have the same mass but different densities. Could these objects ever exert the same pressure?

Step-by-Step Solution

Answer

Explain your reasoning based on the pressure formula. Answer: Yes, it is possible for two objects with the same mass but different densities to exert the same pressure if their heights are equal. This is because if their volumes (V1 and V2) are equal and their heights (h1 and h2) are equal, their areas (A1 and A2) will also be equal. Therefore, when applying the pressure formula (P = F/A), the resulting pressures (P1 and P2) will be equal as well, assuming the same force (mass times gravitational acceleration) acting upon both objects.

Step 1: Analyze the problem using the pressure formula

The pressure formula is P = F/A, where P is the pressure, F is the force, and A is the area. The force acting on an object at rest is equal to the weight of the object, which is the mass (m) times the gravitational acceleration (g). Therefore, F = mg.

Step 2: Calculate the volume of the objects

To determine the area of the objects, we need to know their volumes. Using the density formula, ρ = m/V, we can solve for the volume (V) of each object, as V = m/ρ. Since each object has the same mass, we can set up an equation such that: V1 = m/ρ1 and V2 = m/ρ2 Where V1 and V2 are the volumes of the objects with densities ρ1 and ρ2, respectively.

Step 3: Determine the area of the objects

Assuming the objects are resting on a flat surface, we can calculate the area (A) based on the volume and height. Since A = V/h, we have: A1 = V1/h1 and A2 = V2/h2 If the heights of the objects are equal, then we can rewrite these equations as: A1 = V1/h and A2 = V2/h

Step 4: Compare the pressures of the objects

Now that we have the areas, we can compare the pressures using the formula P = F/A. Substituting F = mg, we have: P1 = mg/A1 and P2 = mg/A2 Since the masses of the objects are the same, we can rewrite these equations as: P1 = \frac{mg}{A1} and P2 = \frac{mg}{A2} To determine if the objects can exert the same pressure, we need to compare these two expressions. If P1 = P2, then: \frac{mg}{A1} = \frac{mg}{A2} Dividing both sides by mg, we get: \frac{1}{A1} = \frac{1}{A2} Substituting the previous equations for A1 and A2 (A1 = V1/h and A2 = V2/h), we get: \frac{1}{V1/h} = \frac{1}{V2/h} Cross-multiplying, we find: V1 * h = V2 * h

Step 5: Conclusion

If two objects with the same mass but different densities have the same volume (V1 = V2) and height (h1 = h2), it is possible for them to exert the same pressure. This is because their areas (A1 and A2) would be equal, resulting in equal pressures (P1 and P2).

Key Concepts

Density

Density is a fundamental property of materials that describes how much mass is contained in a given volume. It is represented by the symbol \(\rho\), and the formula to calculate it is \(\rho = \frac{m}{V}\), where \(m\) is the mass and \(V\) is the volume. This relationship shows that density is directly proportional to mass and inversely proportional to volume. A higher density means that a material has more mass packed into a smaller space.
Understanding density helps us to compare different materials. For instance, a block of iron will have a higher density than a block of wood of the same size, because iron's atoms are packed more tightly.

• Materials with higher density usually feel heavier for the same volume.
• Density determines whether an object will sink or float when placed in a liquid, such as water.

By knowing the density of a material, you can predict its behavior in different situations, such as whether it might exert more or less pressure under the same mass constraints when compared to other materials.

Mass

Mass is a measure of the amount of matter in an object. It is a fundamental concept in physics, often confused with weight, but they are not the same. Weight is the force exerted by gravity on that mass. Mass remains constant regardless of location, whether on Earth, the Moon, or in space.
When comparing objects of different materials but identical mass, as in our exercise, mass plays a key role in how other physical properties, like pressure, are analyzed. The mass (m) of an object can impact:

• How much force an object exerts on a surface it rests on, as seen in the formula \(F = mg\), where \(g\) is the gravitational acceleration.
• How quickly it can move or be accelerated under a given force, represented in Newton's Second Law, \(F = ma\), where \(a\) is acceleration.

Mass is usually measured in kilograms (kg) and does not change even if the density or volume of the material changes. This constancy makes it vital in calculations determining pressure.

Volume Calculation

Calculating volume is essential when we want to understand the space an object occupies. The volume (V) of an object is calculated using its mass and density with the formula \(V = \frac{m}{\rho}\). This means that if two objects have the same mass but different densities, their volumes will differ.
Consider our exercise, where two metals of the same mass have different volumes because of their differing densities. Volume calculation becomes crucial when comparing properties like pressure. Pressure (P) is related to volume through the area on which a force is exerted (\(A\)), as shown in \(P = \frac{F}{A}\), where \(F = mg\).

• Volume impacts how objects interact with their environment, influencing pressure and buoyancy.
• In practical applications, knowing volume helps in designing containers that need to hold specific amounts of substances efficiently.

Volume calculations thus provide insight into material properties and help predict how they will behave in various conditions, crucial for understanding and solving problems involving pressure.