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Magazine Chapter 10: Problem 7
The ratio of coefficients of cubical expansion and linear expansion is (A) \(1: 1\) (B) \(3: 1\) (C) \(2: 1\) (D) None of these
Short Answer
Expert verified
The ratio of coefficients of cubical expansion and linear expansion is not constant and depends on the change in temperature and the cubical coefficient of expansion. Therefore, the correct answer is (D) None of these.
Step by step solution
01
Recall the definitions of coefficients of cubical and linear expansion
The coefficient of linear expansion (\(\alpha\)) is the change in length per unit length for a temperature change of 1掳C. The coefficient of cubical expansion (\(\beta\)) gives the change in volume per unit volume for a temperature change of 1掳C.
02
Coefficient of cubical expansion
Given a substance with a linear, area, and volume increase of 螖L, 螖A, and 螖V, the increase in volume due to temperature change can be obtained by:
\[
\beta = \frac{\Delta V}{V_0\Delta T}
\]
Where \(V_0\) is the initial volume, and \(\Delta T\) is the change in temperature.
03
Relate cubical expansion to linear expansion
When the length, width, and depth of a substance all increase with a linear expansion, the product of these increases is the cubical increase. Let's denote the increase in length, width, and depth as \(\Delta L_1, \Delta L_2,\) and \(\Delta L_3\), respectively. Then the cubical increase is given by:
\[
\Delta V = \Delta L_1 \cdot \Delta L_2 \cdot \Delta L_3
\]
04
Using the coefficients of linear expansion
Using the coefficient of linear expansion \(\alpha\), we can express the increases in length, width, and depth as follows:
\[
\Delta L_1 = \alpha L_1\Delta T
\]
\[
\Delta L_2 = \alpha L_2\Delta T
\]
\[
\Delta L_3 = \alpha L_3\Delta T
\]
Where \(L_1, L_2, L_3\) are the initial length, width, and depth, respectively.
05
Calculate the cubical increase using the coefficients of linear expansion
Now substituting these linear increases into the cubical increase expression, we get:
\[
\Delta V = (\alpha L_1\Delta T) \cdot (\alpha L_2\Delta T) \cdot (\alpha L_3\Delta T)
\]
\[
\Delta V = \alpha^3 L_1L_2L_3 (\Delta T)^3
\]
Now, \(L_1L_2L_3 = V_0\), the initial volume of the substance, so we can rewrite the equation:
\[
\Delta V = \alpha^3 V_0 (\Delta T)^3
\]
06
Relate the cubical increase and the coefficients of linear and cubical expansion
Using the equation for the coefficient of cubical expansion, we can rewrite the equation:
\[
\beta = \frac{\Delta V}{V_0\Delta T}
\]
Substitute the expression for \(\Delta V\):
\[
\beta = \frac{\alpha^3 V_0 (\Delta T)^3}{V_0\Delta T}
\]
This simplifies to:
\[
\beta = \alpha^3 (\Delta T)^2
\]
07
Ratio of coefficients of cubical expansion to linear expansion
Now we want to find the ratio of \(\beta\) to \(\alpha\). Dividing both sides of the equation by \(\alpha^3\), we get:
\[
\frac{\beta}{\alpha^3} = (\Delta T)^2
\]
Divide both sides by \((\Delta T)^2\):
\[
\frac{\beta}{(\Delta T)^2}= \alpha^3
\]
Now take the cube root of both sides of the equation:
\[
\sqrt[3]{\frac{\beta}{(\Delta T)^2}}= \alpha
\]
Now to find the ratio of \(\alpha : \beta\):
\[
\frac{\alpha}{\beta} = \frac{\sqrt[3]{\frac{\beta}{(\Delta T)^2}}}{\beta}
\]
\[
\frac{\alpha}{\beta} = \frac{1}{\beta^{2/3} (\Delta T)^{4/3}}
\]
The ratio of coefficients of cubical expansion and linear expansion is not constant and depends on the change in temperature and the cubical coefficient of expansion. Therefore, the correct answer is:
(D) None of these
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coefficient of Cubical Expansion
In thermal physics, understanding how substances expand or contract with temperature changes is crucial. The coefficient of cubical expansion, denoted as \(\beta\), describes how a material's volume changes with temperature. It is defined as the change in volume per unit volume for every degree change in temperature.
To mathematically express this, the formula is:
To mathematically express this, the formula is:
- \[\beta = \frac{\Delta V}{V_0 \Delta T}\]
Coefficient of Linear Expansion
The coefficient of linear expansion, represented by \(\alpha\), measures how a material's length changes with temperature. This coefficient is defined as the change in length per unit length for each degree of temperature change.
The equation for linear expansion is:
The equation for linear expansion is:
- \[\Delta L = \alpha L_0 \Delta T\]
- \(\Delta L\) is the change in length due to temperature change,
- \(L_0\) is the original length of the material, and
- \(\Delta T\) is the change in temperature.
Thermal Physics
Thermal physics is the field of study that deals with the relationship between heat and other forms of energy. A key aspect of thermal physics is understanding how substances react to changes in temperature, particularly in terms of expansion and contraction.
The concepts of linear and cubical expansions are part of the broader study of thermal physics. They reflect how materials respond to thermal energy, influencing both macroscopic physical changes in materials and microscopic particle dynamics.
The concepts of linear and cubical expansions are part of the broader study of thermal physics. They reflect how materials respond to thermal energy, influencing both macroscopic physical changes in materials and microscopic particle dynamics.
- Temperature affects particle vibrations in solids, leading to expansion.
- Expansion coefficients (linear and cubical) help quantify these changes.
