You are boiling pasta and absentmindedly grab a copper stirring spoon rather than your wooden spoon. The copper spoon has a 20 mm ×1.5 mm rectangular cross section, and the distance from the boiling water to your 35°C hand is 18 cm. How long does it take the spoon to transfer 25 J of energy to your hand?

Verified step by step guidance

1. Step 1: Identify the formula for heat conduction. The rate of heat transfer through a material is given by Fourier's law: \( Q = \frac{k \cdot A \cdot \Delta T \cdot t}{L} \), where \( Q \) is the heat transferred, \( k \) is the thermal conductivity of the material, \( A \) is the cross-sectional area, \( \Delta T \) is the temperature difference, \( t \) is the time, and \( L \) is the length of the material.
2. Step 2: Calculate the cross-sectional area \( A \) of the copper spoon. The cross-section is rectangular, so \( A = \text{width} \times \text{height} = 20 \, \text{mm} \times 1.5 \, \text{mm} = 20 \times 10^{-3} \cdot 1.5 \times 10^{-3} \, \text{m}^2 \).
3. Step 3: Determine the temperature difference \( \Delta T \). The boiling water is at \( 100^{\circ} \text{C} \) and your hand is at \( 35^{\circ} \text{C} \), so \( \Delta T = 100 - 35 = 65^{\circ} \text{C} \).
4. Step 4: Rearrange Fourier's law to solve for time \( t \): \( t = \frac{Q \cdot L}{k \cdot A \cdot \Delta T} \). Substitute the given values: \( Q = 25 \, \text{J} \), \( L = 18 \, \text{cm} = 0.18 \, \text{m} \), \( k \) (thermal conductivity of copper) \( \approx 385 \, \text{W/m·K} \), \( A \) (calculated in Step 2), and \( \Delta T \) (calculated in Step 3).
5. Step 5: Perform the substitution and simplify the expression to find \( t \). Ensure all units are consistent (meters, seconds, joules, etc.) before calculating the time required for the spoon to transfer 25 J of energy to your hand.

Key Concepts

Heat Transfer

Heat transfer is the process by which thermal energy moves from a hotter object to a cooler one. This can occur through conduction, convection, or radiation. In the context of the question, the copper spoon conducts heat from the boiling water to your hand, illustrating conduction, which is the transfer of heat through direct contact between materials.

Thermal Conductivity

Thermal conductivity is a material property that indicates how well a substance can conduct heat. Copper has a high thermal conductivity, meaning it transfers heat efficiently compared to materials like wood. This property is crucial in determining how quickly the spoon will transfer energy to your hand when it comes into contact with the boiling water.

Energy Transfer and Time

The rate of energy transfer is related to both the thermal conductivity of the material and the temperature difference between the two objects. In this scenario, the time it takes for the spoon to transfer 25 J of energy to your hand can be calculated using the formula Q = k * A * (ΔT / d) * t, where Q is the heat transferred, k is the thermal conductivity, A is the cross-sectional area, ΔT is the temperature difference, and d is the distance.