Thermal Expansion Coefficients Relation
The question asks for the relationship between the thermal coefficient of linear expansion (\(\alpha\)) and the thermal coefficient of volume expansion (\(\beta\)) for a material.
Understanding Linear Expansion
The thermal coefficient of linear expansion, denoted by \(\alpha\), describes how the length of a material changes with temperature. For a small change in temperature \(\Delta T\), the change in length \(\Delta L\) is related to the original length \(L_0\) by the formula:
\(\Delta L = \alpha L_0 \Delta T\)
This means the new length \(L\) is:
\(L = L_0 + \Delta L = L_0 + \alpha L_0 \Delta T = L_0 (1 + \alpha \Delta T)\)
Deriving Volume Expansion Coefficient
The thermal coefficient of volume expansion, denoted by \(\beta\), describes how the volume of a material changes with temperature. For a small change in temperature \(\Delta T\), the change in volume \(\Delta V\) is related to the original volume \(V_0\) by:
\(\Delta V = \beta V_0 \Delta T\)
Consider a cube made of the material with initial side length \(L_0\). Its initial volume is \(V_0 = L_0^3\). After a temperature change \(\Delta T\), each side length becomes \(L = L_0(1 + \alpha \Delta T)\).
The new volume \(V\) is:
\(V = L^3 = \left[ L_0 (1 + \alpha \Delta T) \right]^3\)
\(V = L_0^3 (1 + \alpha \Delta T)^3\)
Expanding the term \((1 + \alpha \Delta T)^3\) using the binomial expansion:
\(V = V_0 \left( 1 + 3(\alpha \Delta T) + 3(\alpha \Delta T)^2 + (\alpha \Delta T)^3 \right)\)
In most practical cases, the coefficient of linear expansion \(\alpha\) is small, and the temperature change \(\Delta T\) is moderate. Therefore, the terms involving \((\alpha \Delta T)^2\) and \((\alpha \Delta T)^3\) are very small compared to the other terms and can be neglected.
So, the volume \(V\) can be approximated as:
\(V \approx V_0 (1 + 3\alpha \Delta T)\)
The change in volume is:
\(\Delta V = V - V_0 \approx V_0 (1 + 3\alpha \Delta T) - V_0 = 3\alpha V_0 \Delta T\)
Comparing this approximated expression for \(\Delta V\) with the definition \(\Delta V = \beta V_0 \Delta T\), we find:
\(\beta = 3\alpha\)
Conclusion
The thermal coefficient of volume expansion (\(\beta\)) is three times the thermal coefficient of linear expansion (\(\alpha\)) for an isotropic material.
Similar Questions
Greater the value of _______ of a material, the more rapidly it will conduct heat.
Identify the material having low coefficient of volume expansion
Identify the material having high coefficient of volume expansion.
Identify the material having the lowest coefficient of linear expansion.
- When 1 kg of water is cooled from \(4^{\circ}C\) to \(0^{\circ}C\), its volume _____.
Important Questions from Thermal expansion
Thermal expansion of solids are:
A copper rod and a steel rod are to have lengths LC and LS, such that the difference between their lengths is the same at all ambient temperatures. If the coefficients of linear expansion of copper and steels are αC and αS respectively. The lengths are related to the coefficient of linear expansion as :
How much should the temperature of a brass rod be increased so as to increase its length by 1%?
Given: for brass α = 0.00002/°CA wooden wagon wheel has an outside diameter of 3750 mm. The iron tire for this wheel is deliberately made smaller so that it can be shrunk in place to be a tight fit. If the tire's inside diameter is 3737 mm at 20°C, the temperature to which it must be heated to fit over the wheel? The coefficient of linear expansion of the steel is 1.2 × 10-5/°C.
A cylinder of cross-sectional radius 1 cm and height 4 cm is heated from 0°C to 100°C. If the coefficient of linear expansion α = 4 × 10-4/°C, what will be the increase in the volume of the cylinder?
