the coeffiecient of linear of a crystal in one dirn is a1 and that in every dirm perpendicuar a2 .the coefficient of cubical expansion is:

a)a1+a2 b)2a1+a2 c)a1+2a2 d)none of these

To determine the coefficient of cubical expansion of a crystal when given the linear coefficients in different directions, we need to understand how these coefficients relate to volume changes. The coefficient of linear expansion describes how much a material expands in one dimension per degree change in temperature. When we consider a three-dimensional object, the total volumetric expansion can be derived from the linear expansions in each direction.

Understanding Linear and Volumetric Expansion

The coefficient of linear expansion, denoted as α, indicates how much a unit length of a material expands per degree of temperature change. For a crystal, if we have:

• α1 for the linear expansion in one direction
• α2 for the linear expansion in the two perpendicular directions

When we consider volumetric expansion, we need to account for all three dimensions. The volumetric expansion coefficient (β) can be expressed in terms of the linear coefficients as follows:

Formula for Volumetric Expansion

The relationship between the linear coefficients and the volumetric coefficient is given by:

β = α1 + α2 + α2

This can be simplified to:

β = α1 + 2α2

Analyzing the Options

Now, let's analyze the options provided:

• a) α1 + α2
• b) 2α1 + α2
• c) α1 + 2α2
• d) none of these

From our derived formula, we see that the correct expression for the coefficient of cubical expansion is:

α1 + 2α2

Final Answer

Thus, the correct answer is c) α1 + 2α2. This reflects how the volume of the crystal expands based on its linear expansion in the specified directions.