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Density and Thermal Expansion Relations

The thermal-expansion coefficient (alpha_P) is a thermodynamic quantity defined as

Equation (1)

where P, V, and T are respectively, volume, and temperature. We will refer to (alpha_P) as the instantaneous volumetric thermal-expansion coefficient. For simplicity, the subscript P has been eliminated from the thermal-expansion coefficients in the following discussion with the understanding that constant pressure is implied in all the following equations.

Equation (2)

where V and V₀ are the volumes at temperatures T and T₀ respectively.

Because many measurements of thermal expansion involve measurement of a length change, it is common to find tabulations of the fractional (or percent) change in length,

Equation (3)

where L and L₀ are respectively the sample lengths at temperatures T and T₀.

The instantaneous linear thermal-expansion coefficient is

Equation (4)

The mean linear thermal-expansion coefficient is:

Equation (5)

The instantaneous volumetric thermal-expansion coefficient is just three time the instantaneous linear thermal-expansion coefficient; i.e., alpha = 3 x alpha_l. The same relation does not hold for the mean thermal-expansion coefficients, as the following considerations show. The mean volumetric thermal-expansion coefficient may be written as:

Equation (6)

Since V = L³ the definition of the mean linear thermal-expansion coefficient in Eq. (4) gives

Equation (7)

when this is substituted in Eq. (6) and expanded, the relationship between the mean volumetric and mean linear coefficient is:

Equation (8)

The error introduced by taking only the first term in this equation will generally be small in many applications. For example for Delta T = 1000 and alpha_l = 1 x 10^-5, only a 1% error will be introduced by ignoring the last two terms.

The relation between linear thermal-expansion and density is

Equation (9)

where Delta rho = rho - rho_o is the difference between densities at temperatures T and T₀.

Equation (9) may be derived from the definition of density

Equation (10)

giving

Equation (11)

and the relation between fractional change in volume and fractional change in length

Equation (12)