# Isothermal expansion of a gas: heat of surroundings

santimirandarp
In an isothermal process, for an expanding gas ##\Delta U_{sys}=0## and ##Q=-W## but then,
• How can we evaluate ##Q_{surr} ##?
It should be ##Q_{surr}=-Q_{sys}##, but I don't know how to show it in equations.

If I try to get the result through the principles:

##\Delta U_{sys}=-\Delta U _{surr}=0## but then nothing appears.

Any help?

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Mentor
Heat is not a property of a system, it is a property of a process. Heat is flow of thermal energy, hence the energy leaving leaving the system as heat must end up somewhere, namely the surroundings.

santimirandarp
Heat is not a property of a system, it is a property of a process. Heat is flow of thermal energy, hence the energy leaving leaving the system as heat must end up somewhere, namely the surroundings.
I couldn't derive it from 1st principle.

Mentor
Well, if the heat that passes through the boundary between the system and the surroundings leaves the surroundings and enters the system, the system must be gaining that amount of heat and the surroundings must be losing it. Where else can it be coming from?

Mentor
I couldn't derive it from 1st principle.
If you want to derive it from first principles, consider the combination of system and surroundings as a new isolated system, such that ##Q_{macro}=0##, ##W_{macro}=0##, and ##\Delta U_{macro}=0##. So, $$Q_{system}+Q_{surroundings}=Q_{macro}=0$$
$$W_{system}+W_{surroundings}=W_{macro}=0$$ and $$\Delta U_{system}+\Delta U_{surroundings}=\Delta U_{macro}=0$$

santimirandarp
santimirandarp
Well, if the heat that passes through the boundary between the system and the surroundings leaves the surroundings and enters the system, the system must be gaining that amount of heat and the surroundings must be losing it. Where else can it be coming from?
Thanks, yes but there is something else. Suppose we say '30J are transferred from the system to the environment as heat' it is completely abstract to me. What do we mean by energy here? It's not something easy to picture as work is, it seems. What do we mean by 'energy transferred as heat' that makes it obvious that the system received 30J as heat too?

I think the confusion comes from not having a good picture about what heat is; and looking through the web isn't very helpful so far.

Mentor
Thanks, yes but there is something else. Suppose we say '30J are transferred from the system to the environment as heat' it is completely abstract to me. What do we mean by energy here? It's not something easy to picture as work is, it seems. What do we mean by 'energy transferred as heat' that makes it obvious that the system received 30J as heat too?

I think the confusion comes from not having a good picture about what heat is; and looking through the web isn't very helpful so far.
Are you aware of the physical mechanisms by which heat energy can be transferred across the boundary between a system and its surroundings?

santimirandarp
Are you aware of the physical mechanisms by which heat energy can be transferred across the boundary between a system and its surroundings?
Yes, I am, but still miss something -I'll try to better understand what it is- that makes difficult to see that energy released as heat is absorbed as heat in a system. For example, why isn't this heat diminished by causing some orderly motion in the system?

And also, why isn't it possibly that ##Q_{env}=-(Q+W)_{sys}##?

Mentor
Yes, I am, but still miss something -I'll try to better understand what it is- that makes difficult to see that energy released as heat is absorbed as heat in a system. For example, why isn't this heat diminished by causing some orderly motion in the system?

And also, why isn't it possibly that ##Q_{env}=-(Q+W)_{sys}##?
Do you not remember from mechanics that if A does work on B, B does an equal and opposite amount of work on A?

santimirandarp
Do you not remember from mechanics that if A does work on B, B does an equal and opposite amount of work on A?
I do. But isn't usually assumed that work in the environment is zero, and temperature is constant?

And also, do you mean always ##Q_{env}=-Q_{sys}## and ##W_{sys}=-W_{env}##?

Mentor
I do. But isn't usually assumed that work in the environment is zero
No, the system can do work on the environment, and the environment can do work on the system.
, and temperature is constant?
No. Consider an ice bath that can receive or discharge heat without its temperature changing. Usually, the environment is assumed to have constant temperature in this sense. But, of course, not always.
And also, do you mean always ##Q_{env}=-Q_{sys}## and ##W_{sys}=-W_{env}##?
Yes, for a closed system.