Question

Some textbooks use the unit \(\mathrm{K}^{-1}\) rather than \({ }^{\circ} \mathrm{C}^{-1}\) for values of the linear expansion coefficient; see Table 17.2 How will the numerical values of the coefficient differ if expressed in \(\mathrm{K}^{-1}\) ?

Short answer

Answer: No, there is no difference between the linear expansion coefficient when expressed in K⁻¹ and °C⁻¹. This is because both Kelvin and Celsius scales have the same incremental difference in temperature, which influences the numerical value of the linear expansion coefficient.

Step-by-step solution

Relationship of temperature scales

Both the Celsius (C) and Kelvin (K) scales of temperature have the same incremental difference (1 degree), but their numbering systems start from different points: absolute zero (0 K) for the Kelvin scale and the freezing point of water (0 °C) for the Celsius scale. The relationship between these two scales can be expressed as: T(K) = T(°C) + 273.15

Analyzing the expression of linear expansion coefficient in both units

The linear expansion coefficient, α, describes the tendency of a solid to expand or contract as a function of temperature changes. Mathematically, it is given by: ΔL = L₀αΔT, where ΔL is the change in length, L₀ is the initial length, and ΔT is the change in temperature. Since the incremental difference in temperature for both Kelvin and Celsius scales is the same, we should not observe any significant difference when expressed in K⁻¹ or °C⁻¹. The linear expansion coefficient should remain the same.

Comparing numerical values of the same coefficient in K⁻¹ and °C⁻¹

It can be verified that the numerical values of the coefficients will not differ when expressed in K⁻¹ or °C⁻¹. This is because the main effect of the change of temperature is what influences the numerical value of the linear expansion coefficient. Since the increments are the same in both scales, the values of α in K⁻¹ and °C⁻¹ will be equal for the same temperature change.

Key concepts

Temperature Scales

Understanding different temperature scales is essential in thermodynamics. The Celsius and Kelvin scales are two commonly used temperature measurement systems. These scales increment similarly, meaning a change of 1 degree Celsius is equal to a change of 1 Kelvin.
The difference lies in their starting points:

• Celsius Scale: Begins at the freezing point of water, which is 0 °C. Used widely in most parts of the world for everyday temperature readings.
• Kelvin Scale: Starts at absolute zero (0 K), the point at which absolutely no molecular motion occurs. It is primarily used in scientific contexts.

To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature: \[ T(K) = T(°C) + 273.15 \]
This makes both scales incredibly useful and interchangeable in scientific equations, as they maintain a consistent interval.

Linear Expansion Coefficient

The linear expansion coefficient is a crucial concept in understanding how materials expand or contract with temperature changes. When materials are heated, they tend to expand; conversely, they contract when cooled. This behavior can be quantified using the linear expansion coefficient, usually denoted by \( \alpha \).
Mathematically, the change in length of an object, \( \Delta L \), due to a temperature change, \( \Delta T \), is described by the formula:\[ \Delta L = L_0 \alpha \Delta T \]where \( L_0 \) is the original length of the material.
The linear expansion coefficient is expressed in units of either \( \mathrm{K}^{-1} \) or \( ^{\circ} \mathrm{C}^{-1} \). Because both temperature scales progress with the same incremental change, the numerical value of \( \alpha \) remains unaffected by whether it is expressed per Kelvin or per Celsius degree. This uniformity simplifies calculations and comparisons across various scientific and engineering disciplines.

Kelvin and Celsius Relationship

The connection between Kelvin and Celsius is fundamental in thermal physics and engineering. Since both scales increase at identical rates, measuring temperature changes in degrees Celsius is equivalent to measuring changes in Kelvin. This relationship is beneficial because:

• It ensures compatibility: Scientists and engineers can choose either scale without affecting the outcomes of their calculations.
• The conversion factor is straightforward: By adding 273.15, you convert a temperature from Celsius to Kelvin.

This relationship allows the direct application of thermal expansion equations regardless of the unit used. Therefore, whether you use \( \mathrm{K}^{-1} \) or \( ^{\circ} \mathrm{C}^{-1} \) to express the linear expansion coefficient, the scientific principles and results remain consistent. This unified approach is not only practical but also essential for cross-disciplinary studies and global scientific communication.