Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Two rods, one made of brass and the other made of copper, are joined end to end. The length of the brass section is 0.300 m and the length of the copper section is 0.800 m. Each segment has cross-sectional area 0.00500 m2. The free end of the brass segment is in boiling water and the free end of the copper segment is in an ice–water mixture, in both cases under normal atmospheric pressure. The sides of the rods are insulated so there is no heat loss to the surroundings. (a) What is the temperature of the point where the brass and copper segments are joined? (b) What mass of ice is melted in 5.00 min by the heat conducted by the composite rod?

Short Answer

Expert verified
(a) The joint temperature is approximately 42.03°C. (b) About 9.45 grams of ice melts.

Step by step solution

01

Understand Thermal Conductivity

The problem involves a composite rod with two materials: brass and copper. We need to understand how heat conducts through them. The thermal conductivity equation is: q=KAdTdxwhere q is the heat transfer per unit time, K is the thermal conductivity, A is the cross-sectional area, dT is the temperature difference, and dx is the length.
02

Define Equilibrium Condition

In thermal equilibrium, the heat transferred through both rods is equal, qbrass=qcopper. Set the heat flows through the brass and copper equal to each other:KbrassTboilingTLbrass=KcopperTTiceLcopper where Tboiling=100C, Tice=0C.
03

Use Known Values

Use the given values: - Length of brass, Lbrass=0.300 m- Length of copper, Lcopper=0.800 m- Thermal conductivity: Kbrass=109 W/mC and Kcopper=401 W/mC- Area, A=0.005 m2
04

Solve for Equilibrium Temperature

Plug values into the equation:109100T0.300=401T00.800Solve for T. Rearrange and simplify:109(100T)0.8=401T0.387.2(100T)=120.3T872087.2T=120.3T8720=207.5TT42.03C
05

Calculate Heat Transferred

Use heat flow equation to find total heat transferred in 5 mins:q=KAdTdxtUse copper section for calculation:q=4010.00542.0300.8300q3156.5 J
06

Determine Mass of Ice Melted

Calculate mass of ice melted using heat of fusion (Lf=334,000 J/kg):mice=qLf=3156.5334,0000.00945 kg

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Transfer
Heat transfer is the process by which thermal energy is exchanged between physical systems, depending on the temperature and the medium used. There are multiple modes of heat transfer: conduction, convection, and radiation. In this exercise, we focus on conduction, which is the transfer of heat through a material without the movement of the material itself.
For conduction, the heat transfer rate can be calculated using the formula:
  • q=KAdTdx
  • where q is the heat transfer per unit of time (measured in watts), K is the thermal conductivity of the material, A is the cross-sectional area through which the heat is conducted, dT is the temperature difference across the material, and dx is the thickness or length of the material.
Understanding these components helps us assess how effectively different materials transfer heat. For example, copper and brass have different thermal conductivities, affecting how heat is transferred through each part of the composite rod.
Temperature Equilibrium
Temperature equilibrium is a concept where two connected bodies or parts of a system reach a consistent temperature, stopping any net flow of heat energy between them. When dealing with composite materials like brass and copper rods, as in the exercise, you identify the equilibrium point by ensuring the heat transfer through each section is equal.
In formula terms, achieving temperature equilibrium involves setting the heat flow rates through both materials equal to each other:
  • KbrassTboilingTLbrass=KcopperTTiceLcopper
  • This ensures the heat entering and leaving the joint point is evenly distributed.
The exercise demonstrates this with an equilibrium temperature calculated around 42.03C. This temperature is the point at which heat transfer balances between the hot and cold ends of the rod.
Phase Change - Melting
Phase changes occur when a substance transitions from one state of matter to another due to changes in temperature or pressure. In this exercise, as heat is conducted through the copper rod to melt ice, a phase change from solid to liquid occurs, which is known as melting. The amount of heat required for a phase change is determined by the heat of fusion, specific to the substance.
For ice, the heat of fusion is 334,000 J/kg, which represents the energy required to convert 1 kg of ice at 0°C to 1 kg of water at 0°C. The heat transfer calculated in the exercise helps determine how much ice melts over a certain period:
  • mice=qLf
  • where q is the heat transferred, and Lf is the heat of fusion.
  • For the example given, about 0.00945 kg of ice melts in 5 minutes.
Such calculations are crucial in understanding thermal management in systems involving phase changes.
Metals - Brass and Copper
Metals like brass and copper are widely used in applications involving heat due to their substantial thermal conductivity. Thermal conductivity represents how quickly and efficiently heat passes through a material. In this exercise, brass and copper rods are chosen for their differing heat conducting capabilities, showcasing an important principle in thermal dynamics.
Each metal has distinct physical properties impacting heat transfer:
  • Brass has a thermal conductivity of 109 W/mC, making it a moderate conductor of heat, useful in scenarios needing controlled heat transfer.
  • Copper, with a higher thermal conductivity of 401 W/mC, rapidly transfers heat, making it ideal for quick heat dissipation applications.
Understanding the properties of different metals allows engineers to design systems efficiently, selecting materials best suited for the desired heat transfer outcomes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaporation of sweat is an important mechanism for temperature control in some warm-blooded animals. (a) What mass of water must evaporate from the skin of a 70.0-kg man to cool his body 1.00 C? The heat of vaporization of water at body temperature (37C) is 2.42×106J/kg. The specific heat of a typical human body is 3480 J/kg K (see Exercise 17.25). (b) What volume of water must the man drink to replenish the evaporated water? Compare to the volume of a soft-drink can (355 cm3).

Consider a poor lost soul walking at 5 km/h on a hot day in the desert, wearing only a bathing suit. This person's skin temperature tends to rise due to four mechanisms: (i) energy is generated by metabolic reactions in the body at a rate of 280 W, and almost all of this energy is converted to heat that flows to the skin; (ii) heat is delivered to the skin by convection from the outside air at a rate equal to kAskin(TairTskin), where k is 54 J/h C m2, the exposed skin area Askin is 1.5 m2, the air temperature Tairis 47C, and the skin temperature Tskin is 36C; (iii) the skin absorbs radiant energy from the sun at a rate of 1400 W/m2; (iv) the skin absorbs radiant energy from the environment, which has temperature 47C. (a) Calculate the net rate (in watts) at which the person's skin is heated by all four of these mechanisms. Assume that the emissivity of the skin is e = 1 and that the skin temperature is initially 36C. Which mechanism is the most important? (b) At what rate (in L/h) must perspiration evaporate from this person's skin to maintain a constant skin temperature? (The heat of vaporization of water at 36C is 2.42×106 J/kg.) (c) Suppose instead the person is protected by light-colored clothing (e0) so that the exposed skin area is only 0.45 m2. What rate of perspiration is required now? Discuss the usefulness of the traditional clothing worn by desert peoples.

A 500.0-g chunk of an unknown metal, which has been in boiling water for several minutes, is quickly dropped into an insulating Styrofoam beaker containing 1.00 kg of water at room temperature (20.0C). After waiting and gently stirring for 5.00 minutes, you observe that the water’s temperature has reached a constant value of 22.0C. (a) Assuming that the Styrofoam absorbs a negligibly small amount of heat and that no heat was lost to the surroundings, what is the specific heat of the metal? (b) Which is more useful for storing thermal energy: this metal or an equal weight of water? Explain. (c) If the heat absorbed by the Styrofoam actually is not negligible, how would the specific heat you calculated in part (a) be in error? Would it be too large, too small, or still correct? Explain.

The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume e = 1 for these surfaces. Find the radii of the following stars (assumed to be spherical): (a) Rigel, the bright blue star in the constellation Orion, which radiates energy at a rate of 2.7×1032W and has surface temperature 11,000 K; (b) Procyon B (visible only using a telescope), which radiates energy at a rate of 2.1×1023W and has surface temperature 10,000 K. (c) Compare your answers to the radius of the earth, the radius of the sun, and the distance between the earth and the sun. (Rigel is an example of a supergiant star, and Procyon B is an example of a white dwarf star.)

In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a 200-W electric immersion heater in 0.320 kg of water. (a) How much heat must be added to the water to raise its temperature from 20.0C to 80.0C? (b) How much time is required? Assume that all of the heater’s power goes into heating the water.

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free