At what temperature a gold ring of diameter 6.230 cm be heated so that it can be fitted on a wooden bangle of diameter 6.241 cm? Both the diameters have been measured at room temperature (27°C).
(Given: coefficient of linear thermal expansion of gold αL = 1.4 × 10–5 K–1)
(Given: coefficient of linear thermal expansion of gold αL = 1.4 × 10–5 K–1)
- 125.7°C
- 91.7°C
- 425.7°C
- 152.7°C
The Correct Option is A
Solution and Explanation
To determine the temperature at which a gold ring expands enough to fit on a wooden bangle, we use the formula for linear expansion:
L = L_0 \cdot (1 + \alpha_L \cdot \Delta T)
Where:
- \(L\) is the final length (or diameter in this case) after expansion.
- \(L_0\) is the initial length (or initial diameter).
- \(\alpha_L\) is the coefficient of linear thermal expansion.
- \(\Delta T\) is the change in temperature.
Given:
- Initial diameter of the gold ring, \(L_0 = 6.230 \, \text{cm}\)
- Final diameter needed, \(L = 6.241 \, \text{cm}\)
- \(\alpha_L = 1.4 \times 10^{-5} \, \text{K}^{-1}\)
- Initial room temperature, \(T_0 = 27^\circ \text{C}\)
We need to find the temperature \(T\) such that the expanded diameter equals \(6.241 \, \text{cm}\).
First, we calculate the required change in temperature \(\Delta T\):
Using the formula, set \(L = L_0 (1 + \alpha_L \cdot \Delta T)\) and solve for \(\Delta T\):
6.241 = 6.230 \times (1 + 1.4 \times 10^{-5} \times \Delta T)
Simplifying, we find:
1 + 1.4 \times 10^{-5} \times \Delta T = \frac{6.241}{6.230}
1.4 \times 10^{-5} \times \Delta T = \frac{6.241}{6.230} - 1
\Delta T = \frac{6.241 - 6.230}{6.230 \times 1.4 \times 10^{-5}}
\Delta T \approx \frac{0.011}{6.230 \times 1.4 \times 10^{-5}} \approx 98.7 \, \text{C}
Therefore, the final temperature \(T\) is:
T = T_0 + \Delta T = 27 + 98.7 = 125.7^\circ \text{C}
Thus, the temperature to which the gold ring should be heated is \(125.7^\circ \text{C}\), making the correct answer 125.7°C.
This calculation shows the change in diameter needed to fit the bangle and verifies the application of the linear expansion formula.
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Concepts Used:
Thermal Expansion
Thermal expansion is the tendency of matter to change its shape, area, and volume in response to a change in temperature. Temperature is a monotonic function of the average molecular kinetic energy of a substance.
The expansion of the solid material is taken to be the linear expansion coefficient, as the expansion takes place in terms of height, thickness and length. The gaseous and liquid expansion takes the volume expansion coefficient. Normally, if the material is fluid, we can explain the changes in terms of volume change.
The bonding force among the molecules and atoms differs from material to material. These characteristics of the compounds and elements are known as the expansion coefficient.
