Suppose that the rod in Fig. 17.2417.24a is made of copper, is 45.045.0 cm long, and has a cross-sectional area of 1.251.25 cm2 . Let TH=100.0TH = 100.0°C and TC=0.0TC = 0.0°C. What is the final steady-state temperature gradient along the rod?

Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 17: Temperature and Heat

Problem 57a

Chapter 17, Problem 57a

Suppose that the rod in Fig. $17.24$ a is made of copper, is $45.0$ cm long, and has a cross-sectional area of $1.25$ cm² . Let $T H equals 100.0$ °C and $T C equals 0.0$ °C. What is the final steady-state temperature gradient along the rod?

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Identify the formula for the heat current (H) through a rod, which is given by Fourier's law of heat conduction: H = k * A * (dT/dx), where k is the thermal conductivity, A is the cross-sectional area, and dT/dx is the temperature gradient.

Recognize that in a steady-state condition, the heat current is constant along the rod, and the temperature gradient (dT/dx) can be expressed as (TH - TC) / L, where TH and TC are the temperatures at the hot and cold ends, respectively, and L is the length of the rod.

Substitute the given values into the expression for the temperature gradient: TH = 100.0°C, TC = 0.0°C, and L = 45.0 cm. Convert the length to meters for consistency in units: L = 0.45 m.

The cross-sectional area A is given as 1.25 cm². Convert this to m²: A = 1.25 * 10^-4 m².

Use the known thermal conductivity of copper (k ≈ 385 W/m·K) to calculate the temperature gradient: dT/dx = (TH - TC) / L. This will give you the final steady-state temperature gradient along the rod.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Heat Transfer

Heat transfer is the process by which thermal energy moves from a region of higher temperature to a region of lower temperature. In the context of the rod, heat flows from the hot end (TH) to the cold end (TC) through conduction, which is the transfer of heat through a material without the movement of the material itself.

Thermal Conductivity

Thermal conductivity is a material property that indicates how well a substance can conduct heat. For copper, which has a high thermal conductivity, this means it can efficiently transfer heat along its length. The rate of heat transfer through the rod is directly proportional to the temperature difference and the cross-sectional area, and inversely proportional to the length of the rod.

Temperature Gradient

The temperature gradient is the rate of temperature change with respect to distance in a material. In the case of the copper rod, the temperature gradient can be calculated by dividing the temperature difference (TH - TC) by the length of the rod (L). This gradient is crucial for determining how heat flows through the rod and reaches a steady state.

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