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Thermal conductivity is a fundamental property of materials that indicates how easily heat can flow through them. It is denoted by the symbol \( k \) and measured in units of Watts per meter per Kelvin (W/m·K). Materials with high thermal conductivity, such as metals like copper, can transfer heat very efficiently.
Copper, for instance, has a thermal conductivity around 400 W/m·K. This means that it can quickly transport heat away from a warmer object, like your finger in this exercise, resulting in a cooler touch sensation. On the other hand, glass, with a thermal conductivity of about 1 W/m·K, is much less effective at conducting heat.
This is why, even though two objects are initially at the same temperature, they can feel different to the touch. High thermal conductivity materials remove heat rapidly, giving us the sensation of coolness, while low thermal conductivity materials retain heat near their surface. Thus, understanding thermal conductivity helps us predict how different materials will feel when touched.

Fourier's Law is a cornerstone in the study of heat transfer. It provides a formula to quantify the rate of heat transfer through a material. The law states that the heat transfer rate (\( H \)) depends directly on the thermal conductivity \( k \), the temperature difference between the touching surfaces, and inversely on the thickness of the material.
The mathematical expression for Fourier's Law is given by:

• \( H = k \cdot \frac{T_{finger} - T_{material}}{d} \)

where \( H \) is the heat transfer per unit area, \( T_{finger} \) and \( T_{material} \) are the temperatures of your finger and the material, respectively, and \( d \) is the thickness of the material.
In our exercise, this law helps explain why copper feels cooler than glass. The higher thermal conductivity of copper means that it can quickly transfer heat away from your finger, causing a more pronounced cooling effect. Essentially, Fourier's Law helps us understand how quickly and efficiently different materials can exchange heat.

Thermophysical properties refer to characteristics of materials that define their behavior in response to changes in temperature. These include properties like density \( \rho \), specific heat capacity \( c \), and indeed, thermal conductivity \( k \).

• Density (\( \rho \)) is the mass per unit volume of a material.
• Specific heat capacity (\( c \)) is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius.
• Thermal conductivity (\( k \)) measures a material's ability to conduct heat.

In the context of our exercise, these properties of your finger (\( \rho=1000 \text{ kg/m}^3 \), \( c=4180 \text{ J/kg·K} \), and \( k=0.625 \text{ W/m·K} \)) interact with those of copper and glass to determine the sensation experienced upon touch.
The thermophysical properties can influence how heat moves through a material. For instance, the specific heat capacity affects how much energy the material can absorb before its temperature changes. While these properties are crucial for understanding the full picture of heat transfer, the initial cool touch sensation, as described in this exercise, is more directly influenced by the thermal conductivity of the materials involved.