The ends of a 20-cm-long, 2.0-cm-diameter rod are maintained at 0°C and 100°C by immersion in an ice-water bath and boiling water. Heat is conducted through the rod at 4.5×10^4 J per hour. Of what material is the rod made?

Question

Accepted Answer

Hey, everyone. So this problem is dealing with heat transfer. Let's see what it's asking a metal plate that is centimeters long, a rectangular cross section of three centimeters by two centimeters and its ends are maintained 10 degrees Celsius and 150 degrees Celsius. The rate of heat conduction through this rod is 48.4 times 10 to the four jules per hour. They're asked to determine the material of the metal metal plate. Our multiple choice answers are a aluminum B, stainless steel, C iron or D copper. So the key to solving this problem is we're calling our heat transfer equation. We have Q divided by delta T lower case delta T for our change in time is equal to K, our thermal conductivity multiplied by area multiplied by delta capital T for our change in temperature all divided by D. So they're asking us here to determine the material of the metal plate. The term in this equation that is material dependent is thermal conductivity. So we're going to solve for K and then that will help us identify what material metal we're dealing with here. So, excuse me, so when we rearrange the equation to isolate K, we're left with K equals Q multiplied by D divided by area divided or area multiplied by delta capital T multiplied by delta lowercase T. And so when we look at each of these terms, we have Q divided by delta T together was given to us at the rate of heat conduction. And so we need to get that into standard units. It was given to us as 48.4 times 10 to the four jewels per hour. And we need to get that into jewels per second or watts. And so we're just going to multiply that one hour per 3600 seconds. And that gives us jewels per second. Ok. And so distance we were given in the problem as 25 centimeters, we're gonna rewrite that as 250.25 m. Our area was given to us as three centimeters by two centimeters in standard units. That's six times 10 to the negative four m squared. And then our delta T is the temperature diff difference between either side of the uh plate. So that's 150 centimeters minus 10 centimeters or 140 centimeter sorry degrees Celsius, 150 degrees Celsius minus 10 degrees Celsius equals 148 degrees Celsius. OK. So from here now it is just a plug-in chud to solve for K. So we've got 134 jewels per second multiplied by 0.25 m. All of that divided by our area six times 10 to the negative four m squared multiplied by 140 degrees Celsius. And so our K, when we plug that in, our thermal conductivity comes out to be watts per meter multiplied by degrees Celsius. When we look up our table of thermal conductivity of various metals that aligns with the thermal conductivity for copper. So that is the correct answer for this problem. And that aligns with answer choice D. So that's all we have for this one. We'll see you in the next video.

The ends of a 20-cm-long, 2.0-cm-diameter rod are maintained at 0°C and 100°C by immersion in an ice-water bath and boiling water. Heat is conducted through the rod at 4.5×10^4 J per hour. Of what material is the rod made?

Verified Solution

Key Concepts

Heat Conduction

Thermal Conductivity

Material Properties and Heat Transfer