How Do You Calculate Change in Density Due to Temperature?

The change in density due to temperature can be calculated using the coefficient of cubical expansion. This accounts for the volumetric change of a material as its temperature changes.

Understanding the Formula

The most straightforward method uses this formula:

ρ = ρ_r[1 + b(T − T_r)]

Where:

• ρ is the density at temperature T.
• ρ_r is the known density at a reference temperature T_r.
• b is the coefficient of cubical expansion at the reference temperature and density (ρ_r and T_r). This coefficient represents how much the volume changes per degree of temperature change. It's a material-specific property.
• T is the new temperature.
• T_r is the reference temperature.

This formula directly calculates the density at a new temperature given a known density at a reference temperature and the material's coefficient of cubical expansion.

Another approach calculates the change in density:

Δρ = ρ₀ x α x ΔT

Where:

• Δρ is the change in density.
• ρ₀ is the initial density.
• α is the coefficient of volumetric thermal expansion (similar to b above).
• ΔT is the change in temperature (T - T₀).

This formula provides the change in density which can then be added to or subtracted from the initial density to find the new density. Remember that density decreases with increased temperature for most substances.

Practical Example

Let's say we have water with a density ρ_r = 999.97 kg/m³ at a reference temperature T_r = 4°C. The coefficient of cubical expansion for water around this temperature is approximately b = 2.1 x 10^-4 °C^-1. We want to find the density at T = 20°C.

Using the first formula:

ρ = 999.97 kg/m³ [1 + (2.1 x 10^-4 °C^-1)(20°C - 4°C)]
ρ ≈ 998.2 kg/m³

This shows a decrease in density as expected.

Considerations

• The coefficient of cubical expansion (b or α) is not always constant over a large temperature range. For accurate calculations over significant temperature differences, you may need to use temperature-dependent values of b or α.
• For highly accurate calculations, consider using more advanced thermodynamic models that can account for the complex relationship between density, temperature, and pressure, especially for substances that exhibit unusual behavior near phase transitions (like water near 4°C).