Volumetric Thermal Expansion Calculator
When a substance is heated, its molecules move faster and occupy more space, causing the substance to expand. The volumetric thermal expansion formula dV = beta V dT quantifies this change: the volume change dV equals the volumetric expansion coefficient beta times the initial volume V times the temperature change dT. For solids and liquids, beta is a material property found in thermodynamic tables. For ideal gases, beta = 1/T at constant pressure. This calculation is important in engineering design to prevent failure from thermal stress in pipelines, tanks, and structures, as well as in precision manufacturing and materials science.
Volumetric thermal expansion formula
dV = beta × V × dT
dV is the volume change, beta (1/K) is the volumetric expansion coefficient, V is the initial volume, and dT is the temperature change. The final volume is Vf = V + dV = V(1 + beta dT). For isotropic solids: beta is approximately 3 times the linear thermal expansion coefficient alpha.
Worked example
- A 1,000 L water tank at 20 degrees C is heated by 50 K.
- beta for water = 2.07 x 10^(-4) /K
- dV = 2.07 x 10^(-4) x 1,000 x 50 = 10.35 L
- Final volume = 1,010.35 L
Frequently asked questions
What is volumetric thermal expansion?
Volumetric thermal expansion is the increase in volume of a substance when its temperature increases. The formula is dV = beta V dT, where beta is the volumetric expansion coefficient, V is the initial volume, and dT is the temperature change.
What is the volumetric expansion coefficient beta?
Beta (beta, units 1/K) is the fractional change in volume per degree of temperature change. For ideal gases, beta = 1/T (at constant pressure). For liquids and solids, beta = 3 alpha, where alpha is the linear expansion coefficient.
What are typical volumetric expansion coefficients?
Water at 20 degrees C: beta = 2.07 x 10^(-4) /K. Ethanol: 7.5 x 10^(-4) /K. Steel: 3.6 x 10^(-5) /K (3 x linear). Aluminum: 6.9 x 10^(-5) /K. Air at 20 degrees C (ideal gas): 3.41 x 10^(-3) /K.
Is this the same as linear thermal expansion?
No. Linear expansion gives the change in one dimension: dL = alpha L dT. Volumetric expansion covers all three dimensions: for isotropic materials, beta is approximately 3 alpha. For anisotropic materials (like wood), the expansion differs by direction.
Where is volumetric thermal expansion important?
Engineering applications include pipeline design (expansion joints), tank storage (thermal contraction of liquids), bridge construction, thermometer design, and glass sealing. Water's anomalous contraction from 0 to 4 degrees C is important in ecology and limnology.
Official sources
- NIST Chemistry WebBook: Thermophysical Properties of Fluid Systems.
- NIST: NIST Standard Reference Database 69.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.