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How to calculate coefficient of linear expansion

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The coefficient of linear expansion, often denoted as \( \alpha \), is a material constant that quantifies how much a material expands per unit length for each degree of temperature increase. It is defined by the equation:

\[ \alpha = \frac{1}{L_0} \frac{\Delta L}{\Delta T} \]

where:

  • \( \alpha \) is the coefficient of linear expansion (in units of \( \text{°C}^{-1} \) or \( \text{K}^{-1} \)),
  • \( L_0 \) is the original length of the material,
  • \( \Delta L \) is the change in length,
  • \( \Delta T \) is the change in temperature.

Steps to Calculate the Coefficient of Linear Expansion:

  1. Measure the Original Length:

    • Record the initial length \( L_0 \) of the material at a known temperature.

  2. Change the Temperature:

    • Heat or cool the material to a new temperature, allowing it to reach thermal equilibrium.

  3. Measure the New Length:

    • Measure the new length \( L \) of the material after the temperature change.

  4. Calculate the Change in Length:

    • Compute the change in length \( \Delta L \): \[ \Delta L = L - L_0 \]

  5. Determine the Change in Temperature:

    • Calculate the change in temperature \( \Delta T \): \[ \Delta T = T_{\text{final}} - T_{\text{initial}} \]

  6. Substitute Values into the Coefficient Formula:

    • Finally, substitute \( \Delta L \), \( \Delta T \), and \( L_0 \) into the formula for \( \alpha \): \[ \alpha = \frac{\Delta L}{L_0 \cdot \Delta T} \]

Example Calculation:

  • Suppose you have a metal rod that is originally 1.0 m long at 20 °C.
  • Upon heating it to 100 °C, the new length is measured to be 1.002 m.
  1. \( L_0 = 1.0 , \text{m} \)
  2. \( L = 1.002 , \text{m} \)
  3. \( \Delta L = 1.002 , \text{m} - 1.0 , \text{m} = 0.002 , \text{m} \)
  4. \( T_{\text{initial}} = 20 , \text{°C} \)
  5. \( T_{\text{final}} = 100 , \text{°C} \)
  6. \( \Delta T = 100 - 20 = 80 , \text{°C} \)

Now, calculate \( \alpha \):

\[ \alpha = \frac{0.002 , \text{m}}{1.0 , \text{m} \cdot 80 , \text{°C}} = \frac{0.002}{80} = 0.000025 , \text{°C}^{-1} \text{ or } 2.5 \times 10^{-5} , \text{°C}^{-1} \]

This is the coefficient of linear expansion for the metal rod.