Density is mass divided by volume. If the volume V is temperature dependent, so is the density ρ is show that the change in density ρ. Show that the change in density ∆ρ with change in temperature ∆T is given by ∆ρ = - βρ ∆T, where β is the coefficient of volume expansion. Explain the minus sign

To understand how density changes with temperature, we need to delve into the relationship between mass, volume, and temperature. Density (ρ) is defined as mass (m) divided by volume (V), expressed mathematically as ρ = m/V. When the temperature of a substance changes, its volume typically changes as well, which in turn affects its density. Let's break this down step by step.

Understanding Volume Expansion

Most materials expand when heated and contract when cooled. This behavior is quantified by the coefficient of volume expansion (β), which indicates how much the volume of a substance changes per degree change in temperature. The relationship can be expressed as:

• ΔV = βV₀ΔT

Here, ΔV is the change in volume, V₀ is the original volume, and ΔT is the change in temperature. This equation tells us that for a given initial volume, the change in volume is directly proportional to the temperature change.

Impact on Density

Since density is mass divided by volume, we can express the new density (ρ') after a temperature change as:

• ρ' = m / V'

Where V' is the new volume after the temperature change. We can substitute V' using the volume expansion formula:

• V' = V₀ + ΔV = V₀ + βV₀ΔT = V₀(1 + βΔT)

Now, substituting this back into the density equation gives us:

• ρ' = m / [V₀(1 + βΔT)]

Relating New Density to Original Density

Since the mass (m) remains constant during the temperature change, we can express the new density in terms of the original density:

• ρ' = ρ₀ / (1 + βΔT)

Where ρ₀ is the original density. To find the change in density (Δρ), we can express it as:

• Δρ = ρ' - ρ₀

Calculating the Change in Density

Substituting our expression for ρ' into this equation gives:

• Δρ = (ρ₀ / (1 + βΔT)) - ρ₀

To simplify this, we can factor out ρ₀:

• Δρ = ρ₀ [1/(1 + βΔT) - 1]

Using the approximation for small changes (when βΔT is small), we can use the first-order Taylor expansion:

• 1/(1 + x) ≈ 1 - x

Applying this approximation gives us:

• Δρ ≈ ρ₀ [-βΔT]

Understanding the Negative Sign

The negative sign in the equation Δρ = -βρ₀ΔT indicates that as temperature increases (ΔT > 0), the density decreases (Δρ < 0). This is a direct consequence of the expansion of volume. When a substance expands, its volume increases, leading to a decrease in density since mass remains constant. Conversely, if the temperature decreases, the volume contracts, resulting in an increase in density.

In summary, the relationship between temperature and density is crucial in understanding material behavior. The coefficient of volume expansion (β) plays a key role in quantifying how density changes with temperature, and the negative sign reflects the inverse relationship between temperature and density in most materials.