Ch. 19 - Heat and the First Law of Thermodynamics

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 19 - Heat and the First Law of Thermodynamics

Problem 71

Chapter 19, Problem 71

A copper rod and an aluminum rod of the same length and cross-sectional area are attached end to end (Fig. 19–35). The copper end is placed in a furnace maintained at a constant temperature of 205°C. The aluminum end is placed in an ice bath held at a constant temperature of 0.0°C. Calculate the temperature at the point where the two rods are joined.

Verified step by step guidance

Understand the problem: Heat flows through the copper and aluminum rods until thermal equilibrium is reached at the junction. The heat transfer rate through both rods must be equal at the junction since the system is in steady state.

Write the heat transfer equation for each rod using Fourier's law of heat conduction: \( Q/t = k \cdot A \cdot (T_{hot} - T_{junction}) / L \) for the copper rod and \( Q/t = k \cdot A \cdot (T_{junction} - T_{cold}) / L \) for the aluminum rod, where \( k \) is the thermal conductivity, \( A \) is the cross-sectional area, \( L \) is the length, and \( T \) represents temperature.

Set the heat transfer rates equal to each other because the heat flow through the copper rod equals the heat flow through the aluminum rod at the junction: \( k_{Cu} \cdot (T_{hot} - T_{junction}) = k_{Al} \cdot (T_{junction} - T_{cold}) \).

Rearrange the equation to solve for \( T_{junction} \): \( T_{junction} = \frac{k_{Cu} \cdot T_{hot} + k_{Al} \cdot T_{cold}}{k_{Cu} + k_{Al}} \).

Substitute the given values: \( T_{hot} = 205 \ \degree C \), \( T_{cold} = 0 \ \degree C \), \( k_{Cu} = 385 \ \text{W/m·K} \), and \( k_{Al} = 205 \ \text{W/m·K} \). Simplify the expression to find \( T_{junction} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Thermal Conductivity

Thermal conductivity is a material property that indicates how well a substance can conduct heat. It is defined as the amount of heat that passes through a unit area of the material per unit time for a temperature difference of one degree. Different materials have different thermal conductivities; for instance, copper has a higher thermal conductivity than aluminum, meaning it can transfer heat more efficiently.

Heat Transfer

Heat transfer is the process by which thermal energy moves from one object or substance to another due to a temperature difference. In this scenario, heat flows from the hot end of the copper rod to the cold end of the aluminum rod. The rate of heat transfer depends on the thermal conductivities of the materials and the temperature gradient between them.

Steady-State Condition

A steady-state condition occurs when the temperature at any point in a system does not change over time, despite the ongoing heat transfer. In this problem, once the system reaches steady-state, the temperature at the junction of the copper and aluminum rods will stabilize, allowing us to calculate the temperature at that point based on the heat conducted through each rod.

Recommended video:

Guided course

06:27

Solving Ideal Gas Problems With Changing States

Related Practice

Textbook Question

(a) Estimate the total power radiated into space by the Sun, assuming it to be a perfect emitter at T = 5500 K. The Sun’s radius is 7.0 x 10⁸ m.

(b) From this, determine the power per unit area arriving at the Earth, 1.5 x 10¹¹ m away (Fig. 19–37).

views

Textbook Question

What will be the final result when equal masses of ice at 0°C and steam at 100°C are mixed together?

views

Textbook Question

Show, using Eqs. 19–7 and 19–16, that the work done by a gas that slowly expands adiabatically from pressure P₁ and volume V₁ , to P₂ and V₂, is given by W = (P₁V₁ - P₂V₂) / (γ - 1).

views

Textbook Question

A microwave oven is used to heat 250 g of water. On its maximum setting, the oven can raise the temperature of the liquid water from 20°C to 100°C in 1 min 45 s ( = 105 s).

(a) At what rate does the oven put energy into the liquid water?

(b) If the power input from the oven to the water remains constant, determine how many grams of water will boil away if the oven is operated for 2 min (rather than just 1 min 45 s).

views

Textbook Question

If a heater supplies 1.8 x 10⁶ J/h to a room 3.5 m x 4.6 m x 3.0 m containing air at 20°C and 1.0 atm, by how much will the temperature rise in one hour, assuming no losses of heat or air mass to the outside? Assume air is an ideal diatomic gas with molecular mass 29.

views

Textbook Question

A ceramic teapot (e = 0.70) and a shiny metal one (e = 0.10) each hold 0.55 L of tea at 85°C. (a) Estimate the rate of heat loss from each, and (b) estimate the temperature drop after 30 min for each. Consider only radiation, and assume the surroundings are at 20°C.

views