The volume expansivity βvalues of copper at 300 K and 500 K are 49.2 ×10^-6 K^-1 and 54.2 ×10^-6

step 1 For this problem on the topic of thermodynamic property relations, we are given the volume expan

The volume expansivity βvalues of copper at 300 K and 500 K...

The volume expansivity $\beta$ values of copper at $300 \mathrm{K}$ and $500 \mathrm{K}$ are $49.2 \times 10^{-6} \mathrm{K}^{-1}$ and $54.2 \times 10^{-6} \mathrm{K}^{-1},$ respectively, and $\beta$ varies almost linearly in this temperature range. Determine the percent change in the volume of a copper block as it is heated from $300 \mathrm{K}$ to $500 \mathrm{K}$ at atmospheric pressure.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of matter to change its dimensions (length, area, or volume) in response to a change in temperature. When a material is heated, its particles move more vigorously, which increases the average separation between them and leads to an overall expansion. This concept is fundamental in understanding how materials behave under temperature changes and is crucial in applications that require precise dimensional control across varying temperatures.

Volume Coefficient of Thermal Expansion

The volume coefficient of thermal expansion, often denoted by beta (?), is a parameter that quantifies the fractional change in a material's volume per unit change in temperature at constant pressure. It is defined mathematically as ? = (1/V)(dV/dT). This coefficient is material-specific and plays a critical role in predicting the volumetric changes that occur during heating or cooling.

Linear Approximation of Variable Coefficients

When the coefficient of thermal expansion varies with temperature, it is common to approximate the effect over a temperature range by assuming a linear variation. This allows for the calculation of an average coefficient, which can then be multiplied by the temperature change to estimate the overall fractional change in volume. This method simplifies the analysis when the variation in the coefficient is small or approximately linear over the temperature range considered.

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