# I Temperature of two bodies in the same room and under the Sun

#### fog37

Hello Forum

There are three different mechanisms to transfer thermal energy: conduction, convection and radiation.
It is well known that when two objects having different temperatures T1 and T2 are in contact, they will eventually reach a common intermediate temperature (the weighted average of T1 and T2).
Will different objects have a common and intermediate equilibrium temperature also when the transfer mechanisms are convective and radiative instead of conduction?

When an object is left under the sun for a while, thermal equilibrium will be reached (absorbed energy = emitted energy). Different objects with different emissivity will also reach their own thermal equilibrium and their different final temperatures. The two different objects will also radiate and absorb energy from each other but the main driving thermal source is the Sun. The sun is also in thermal equilibrium at a temperature of 5777 K. The objects left in the sun do NOT reach the same equilibrium temperature as the sun. However, objects in the same room, during daylight, will reach a common temperature. Why does that happen and why do the objects not reach different final temperature?

I guess that objects that are in physical contact, either in the room or under the sun, will always reach the same final equilibrium temperature. But if object not in physical contact under the sun will always have different equilibrium temperature...

Thanks,
Fog37

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#### Dale

Mentor
The sun is also in thermal equilibrium at a temperature of 5777 K. The objects left in the sun do NOT reach the same equilibrium temperature as the sun
Objects which were in thermal contact only with the sun would indeed reach the equilibrium temperature of the sun.

However, objects are generally in thermal contact with some other objects as well. These other heat exchanges, as well as internal thermal transport, usually play a large role in determining the equilibrium temperature.

#### fog37

Thanks.

So, I would conclude, that the an identical equilibrium temperature is only reached for objects that are in physical contact and through conduction.

#### Dale

Mentor
So, I would conclude, that the an identical equilibrium temperature is only reached for objects that are in physical contact and through conduction.
The mechanism doesn't matter much. It can be conduction, convection, or radiation. Any two objects which are only in thermal contact with each other will reach the same equilibrium temperature as each other.

#### fog37

Thanks again.

I guess I am also thinking about the situation of a bathroom in the morning where objects, regardless of being made of different materials, are at the same temperature. Once we touch them (conduction), some objects will "feel" colder and some hotter due to their different thermal conductivities. But their temperatures are the same. I guess enough time went by that thermal equilibrium brought all objects at the same temperature. The same should happen for object in the shadow instead of in the sun: they should all have the same temperature regardless of their different composition...

#### Cutter Ketch

I will try to support what Dale has said. The mechanism of heat transport does not matter. Objects in connection only through radiation will nevertheless reach equilibrium just the same as conduction and convection (well the rates are different, but you know what I mean). The reason everything isn't at the same temperature is because the heat transfer is never between just two objects. Even if I had a universe made up of one sun and one planet, they aren't isolated because they both radiate into the vacuum of space. (This brings up an interesting philosophy discussion as to whether an isolated object can radiate heat into space if there is literally nothing else in the universe. Table that for now. The universe exists.)

This is very nearly the case for the sun and Mercury. Why isn't the surface of Mercury the temperature of the sun? Mercury exchanges heat with the sun which takes up only a small part of its sky. The rest of the sky is cold space. Mercury's temperature is the equilibrium between absorbed radiation from the relatively small disk of the sun and emitted radiation into the large nearly complete sphere of cold space. Put the Sun and Mercury inside a styrofoam box and they will reach a single temperature (as will the interior wall of the box.)

This is true for all of the planets, however the spin and atmosphere of Earth do a much better job of distributing the heat around the planet.

If there were only two objects, emissivity wouldn't matter. It affects the rate, but not the direction of heat flow. Emmisivity would change how long it took to get to equilibrium, but not the value. However as soon as you have a third reservoir (including the cold universe) the rates and therefore the emmisivities do play a role in determining the equilibrium values.

Anything here on Earth is much more complicated because there are heat transfer mechanisms connecting everything. It's very hard to do proper calorimetry. Heat always sneaks in or out from somewhere. Just ask Pons and Fleischmann!

#### fog37

If there were only two objects, emissivity wouldn't matter. It affects the rate, but not the direction of heat flow. Emmisivity would change how long it took to get to equilibrium, but not the value. However as soon as you have a third reservoir (including the cold universe) the rates and therefore the emissivities do play a role in determining the equilibrium values.
Hello mike.albert99. Thanks for your help.

I think that the emissivity has an effect on the final equilibrium temperature reached by a specific object. For instance, in the sun, a black colored object will reach a higher final equilibrium temperature than a white colored object which will reach its lower final equilibrium temperature much more quickly.

I guess I am trying to understand all the circumstances in which two objects at different temperature will reach the same equilibrium T. This result is always presented when two objects at different T are placed into contact (conduction). The sun may not play a role in that case. the environment (surrounding air) may not play a big role either. Why do objects in a room, away from strong sources of radiation, seem to reach the same equilibrium T?

#### Cutter Ketch

Hello mike.albert99. Thanks for your help.

I think that the emissivity has an effect on the final equilibrium temperature reached by a specific object. For instance, in the sun, a black colored object will reach a higher final equilibrium temperature than a white colored object which will reach its lower final equilibrium temperature much more quickly.

I guess I am trying to understand all the circumstances in which two objects at different temperature will reach the same equilibrium T. This result is always presented when two objects at different T are placed into contact (conduction). The sun may not play a role in that case. the environment (surrounding air) may not play a big role either. Why do objects in a room, away from strong sources of radiation, seem to reach the same equilibrium T?
Yes emissivity matters. That's what I was saying. It wouldn't matter if two objects were the only two objects in the universe. In that case they also arrive at the same temperature. However as soon as you have another reservoir equal temps are not required and the rates (and so the emissivities) show up in the equilibrium values. If you put your black and white objects by themselves (individually) into a box with walls at a fixed temperature they will both achieve the same equilibrium temperature, the temperature of the walls, regardless of emissivity. If you put them in the box together they will arrive at different equilibrium temperature. This is particularly pronounced if the box is a vacuum chamber so radiation is the only connection.

As for objects in a room, the answer is convection as the vacuum vessel example above suggests. ( could say conduction but conduction around a typical room is terrible)

#### Dale

Mentor
I think that the emissivity has an effect on the final equilibrium temperature reached by a specific object.
Not if there are only two objects in thermal contact. Then the final equilibrium temperature will not depend on the emissivity. Once there are more objects then it becomes more complicated, without a simple answer.

You may be thinking of the rates of heat transfer, where conduction and convection are proportional to the difference in temperature, but radiation is proportional to the difference in the fourth power of the temperature. So for small differences in temperature the conduction and convection are much faster.

#### Bystander

Homework Helper
Gold Member
final equilibrium temperature
Could we please agree to use "steady state" temperature rather than "equilibrium?"

#### fog37

Bystander,

You are right. The objects will reach steady state, i.e. they will emit as much energy as they absorbs. Thermal equilibrium means that they all reach the same final temperature...

"Temperature of two bodies in the same room and under the Sun"

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