If they both take up heat at the same rate, which will warm to your body temperature first?

Question

You hold a gram of copper in one hand and a gram of aluminum in the other. Each metal was originally at 0oC. (Both metals are in the shape of a little ball that fits into your hand.) If they both take up heat at the same rate, which will warm to your body temperature first?

Answer 1

It would take less time for the Copper metal to warm to your hand's temperature due to its lower specific heat than Aluminum metal.

Q = mc(delta T)
Where m is the mass of the metal, c is the specific heat, and (delta T) is the change in temperature the metal experiences.
Q is the amount of heat energy which would need to be absorbed to cause the increase in temperature.

In this question both the mass and the change in temperature are the same for both samples. Additionally, we are told that the objects absorb heat at the same rate (their 'power' is the same).

The specific heat of Copper metal is: 3.87 E2 J / kg degree C
The specific heat of Aluminum metal is: 8.99 E2 J / kg degree C

Since Copper has a smaller specific heat than Aluminum, it will require less energy to raise its temperature (and everything else in the equation is held constant), so the Copper's temperature will rise more quickly.


Q_Copper = (3870 J/g degree C)(x g degree C) = x * 3870 Joules
Q_Aluminum = (8990 J/g degree C)(x g degree C) = x * 8990 Joules

Power = Energy / time
Time = Energy / Power

Time_Copper = x * 3870 J / Power
Time_Aluminum = x * 8990 J / Power

Since in both cases, x and Power are equal for both Copper and Aluminum, the time it takes the Copper to warm to your hand’s temperature is less than that of Aluminum.

x * 3870 J / Power < x * 8990 J / Power
Time_Copper < Time_Aluminum

Answer 2

There are a few materials that are transparent (or weakly absorbing) to infrared radiation in the 700 cm-1 (14 micron) region that is significant for the greenhouse effect. These include AgCl, KBr and polyethylene. Most other materials absorb quite strongly in this region. Water, which covers most of the earth's surface absorbs all of the incident infrared radiation in this region in less than 10 microns (0.01 mm). The earth's surface approximates a black body, but the emissivity is not quite 1, so the energy radiated is not quite sigma*T^4. Further the emissivity is wavelength dependent. Most infrared radiation is not reflected back to the atmosphere: it is absorbed warming the surface and then re-radiated (read about the Einstein A and B coefficients) with an intensity dependent on the surface temperature (and over a much wider bandwidth). By the Krammers-Kronig relation, there is a large change in the refractive index near strong absorptions that can give rise to narrow spectral regions of high infrared reflectivity. I have used this effect to probe lipid content wet biological samples. The earth responds to increased inward radiation from CO2 by warming enough to radiate the same amount of energy to space. A new equilibrium at a higher temperature is established. I have explained the surface balance only. A full model will treat the atmosphere above as a large number of layers and calculate the radiative transfer between each successive layer. The process of photon diffusion in the atmosphere is not unlike the diffusion of photons through your flesh (although the length scales are very different due to the large density difference). You can shine a light through your hand, but probably not a thicker leg. Observe that the incident light loses directionality. Your flesh will glow homogeneously in regions adjacent to the illumination (decaying exponentially with distance from the point of illumination). Adding CO2 to the atmosphere has the same effect as adding more flesh thickness in the analogy above.

Answer 3

copper will warm up first even though they "take up heat at the same rate". The prickly issue is what is meant by "take up heat" you could read it to mean their temperature will rise at the same rate (resulting in a tie) or that they absorb heat energy at the same rate thus Cu warms up first. The structure of the metal is what determines heat capacity. WIth a much heavier nucleus copper has fewer atoms in a gram than Aluminum hence its easier to get them to vibrate (which is heat, don't you know). Per atom, copper takes much more energy, but per gram, aluminum does.
as far as One WHo Knows. Sorry, he is assuming that the conductivity is the rate limiting parameter rather than, say, the warmth of the hand (blood circulation). He is right that if everything else were equal (in an identical environment) the one with the largst conductivity would heat up faster. But the problem says they heat "at the same rate" hence the environment can't be the same because their conductivities ARE different.

Answer 4

This is a good example of the risk of not doing one's science and in asking for answers on a public forum like this! Sorry, no offence meant.

The problem needs to be approached a little differently. They both can't take up heat at the same rate. Why, their thermal conductivities are different. And since their specific heats are also different, that adds another variable,

Off hand I don't have the data of thermal conductivity and specific heat for both the metals and so cann't give you answer, but I hope you understand how to approach the question.

Answer 5

The copper will heat up to body temperature first. Aluminium has more than twice the specific heat capacity of copper, so it takes more than twice as much heat to heat it up as it does to heat the copper. Since they are both taking in heat at the same rate, the temperature of the copper will rise at about twice the rate.

Aluminium 0.9 Joules per gram per degree rise (Celsius or Kelvin)

Copper 0.4 Joules per gram per degree rise.

Answer 6

You answered your own question here. If you start with the same amount of 2 different metals (1 gram each) and each starts out at the same temperature and each absorbs heat at the same rate, then both will warm up at the same rate and reach the temperature in question at the same time.

Answer 7

Neither (or both), you have the same amount to heat up and they take up heat at the same rate, they will become body temperature at the same time.

Answer 8

I am purely guessing here, but I think that the copper would due to its nature of being more susceptible to corrosion and chemicals - the oils in your skin would likely cause a reaction which would make for a quicker heat transfer. Let me know if I Am right.

Answer 9

well seeing as the question states at the same rate then both will be heatd equaly.