Question

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1) The volume thermal expansion coefficient is defined as the fractional change in volume of a substance per unit change in temperature. Consider a closed chamber containing a monatomic ideal gas at atmospheric pressure. Consider its behavior as its temperature is increased by a small amount T, and find its volume expansion coefficient, assuming (a) isobaric and (b) adiabatic conditions. Finally, (c) discuss why these two values are different, and compare them to expansion coefficients of liquids and solids.

Answer 1

Answer 2

Temperature is defined as the average molecular kinetic energy of a substance. When a substance is heated, the kinetic energy of its molecules increases. Thus, the molecules begin moving more and usually maintain a greater average separation. Materials which contract with increasing temperature are unusual; this effect is limited in size, and only occurs within limited temperature ranges (see examples below). The degree of expansion divided by the change in temperature is called the material's coefficient of thermal expansion and generally varies with temperature.

For solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain, given by and defined as:

thermal

finalJinitial €thermal- Linitial

where
Linitial
is the length before the change of temperature and
Lfinal
is the length after the change of temperature.

For most solids, thermal expansion is proportional to the change in temperature:

€thermal X ДТ

Thus, the change in either the strain or temperature can be estimated by:

€thermal-QL Δ7

where

-I initial

For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient:

ау

This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, for small differential changes, one-third of the volumetric expansion is in a single axis. As an example, take a cube of steel that has sides of length L. The original volume will be $V=L^3$ and the new volume, after a temperature increase, will be

AL

We can make the substitutions $\Delta V=\alpha_V L^3\Delta T$ and, for isotropic materials, $\Delta L=\alpha_L L \Delta T$ . We now have:

3 2

Since the volumetric and linear coefficients are defined only for extremely small temperature and dimensional changes (that is, when $\Delta T$ and $\Delta L$ are small), the last two terms can be ignored and we get the above relationship between the two coefficients. If we are trying to go back and forth between volumetric and linear coefficients using larger values of $\Delta T$ then we will need to take into account the third term, and sometimes even the fourth term.

The thermal expansivity of a mixture from the expansivities of the pure components and their excess expansivities follow from:

$\frac{\partial V}{\partial T} = \sum_i \frac{\partial V_i}{\partial T} + \sum_i \frac{\partial V_i^{E}}{\partial T}$

$\alpha= \sum_i \alpha_i V_i + \sum_i \alpha_i^{E} V_i^{E}$

$\frac{\partial \bar{V^E}_i}{\partial T} = R \frac{\partial (ln(\gamma_i))}{\partial P} +RT {\partial^2\over\partial T\partial P} ln(\gamma_i)$