A brass cube has an edge length of 33.2 cm at 20.0°C. Find (a) the increase in surface area and (b) the increase in volume when it is heated to 75.0°C.

To solve the problem of finding the increase in surface area and volume of a brass cube when heated from 20.0°C to 75.0°C, we need to consider how temperature affects the dimensions of solids. Brass, like most materials, expands when heated. The change in dimensions can be calculated using the coefficients of linear expansion.

Understanding Linear Expansion

The linear expansion of a material is described by the formula:

ΔL = L₀ * α * ΔT

• ΔL is the change in length.
• L₀ is the original length (edge length of the cube).
• α is the coefficient of linear expansion for the material (for brass, it's approximately 19 x 10^-6 °C^-1).
• ΔT is the change in temperature.

Calculating the Change in Length

First, we need to determine the change in temperature:

ΔT = 75.0°C - 20.0°C = 55.0°C

Now we can calculate the change in length:

ΔL = 33.2 cm * (19 x 10^-6 °C^-1) * 55.0°C

Calculating this gives:

ΔL ≈ 33.2 cm * 1.045 x 10^-3 ≈ 0.0347 cm

New Edge Length

The new edge length after heating is:

L = L₀ + ΔL = 33.2 cm + 0.0347 cm ≈ 33.2347 cm

Surface Area Increase

The surface area (A) of a cube is calculated using the formula:

A = 6 * L²

Calculating the original surface area:

A₀ = 6 * (33.2 cm)² ≈ 6 * 1102.24 cm² ≈ 6613.44 cm²

Now, calculating the new surface area:

A = 6 * (33.2347 cm)² ≈ 6 * 1105.56 cm² ≈ 6633.36 cm²

The increase in surface area is:

ΔA = A - A₀ = 6633.36 cm² - 6613.44 cm² ≈ 19.92 cm²

Volume Increase

The volume (V) of a cube is given by:

V = L³

Calculating the original volume:

V₀ = (33.2 cm)³ ≈ 36722.88 cm³

Now, calculating the new volume:

V = (33.2347 cm)³ ≈ 36757.66 cm³

The increase in volume is:

ΔV = V - V₀ = 36757.66 cm³ - 36722.88 cm³ ≈ 34.78 cm³

Summary of Results

In summary, when the brass cube is heated from 20.0°C to 75.0°C:

• The increase in surface area is approximately 19.92 cm².
• The increase in volume is approximately 34.78 cm³.

This demonstrates how thermal expansion affects both the surface area and volume of materials, which is an important concept in physics and engineering. Understanding these changes is crucial for applications where temperature variations are expected.