Question

So, this has been bugging me for a while.

Consider the isothermal expansion of an ideal gas in a container. As the gas is allowed to expand, it does work on the surroundings which causes the temperature of the gas to drop. Since it is isothermal expansion, energy must flow via heat into the gas from the surroundings to maintain the temperature. The total change in entropy is given by

dSₜₒₜₐₗ = dSₛ_yₛₜₑₘ + dSₛᵤᵣᵣₒᵤₙ_dᵢₙ_gₛ

Now, the entropy change for the system is

dSₛ_yₛₜₑₘ = dSₚᵣₒ_d_cₑ_d + dSₑₓ_cₕₐₙ_gₑ_d

where, dSproduced is internally produced entropy in the system and dSexchanged is entropy change due to exchange of energy via heat.

Let the amount of heat exchanged be
δqₛ_yₛₜₑₘ = −δqₛᵤᵣᵣₒᵤₙ_dᵢₙ_gₛ,
so dSₑₓ_cₕₐₙ_gₑ_d over the temperature T is
δqₛ_yₛₜₑₘ_ / T = −δqₛᵤᵣᵣₒᵤₙ_dᵢₙ_gₛ / T

Then,
dSₜₒₜₐₗ = dSₛ_yₛₜₑₘ + dSₛᵤᵣᵣₒ-uₙ_dᵢₙ_gₛ = d S ₚᵣₒ_dᵤ_cₑ_d + δqₛ_yₛₜₑₘ_ / T − δqₛᵤᵣᵣₒᵤₙ_dᵢₙ_gₛ / T = dSₚᵣₒ_dᵤ_cₑ_d

My question is that why should we account for the energy given via heat by surroundings which is δqₛ_yₛₜₑₘ / T
, to the system, as it is already used by the gas to maintain the temperature. For an irreversible isothermal expansion the internally produced entropy will be greater than zero due to increase in volume of the gas. But how will heat exchanged add to total entropy when it is only used to maintain the temperature of the system.

Answer 1

In an isothermal expansion of an ideal gas, heat is exchanged with the surroundings to maintain the temperature, resulting in an increase in entropy. The total change in entropy includes both internally produced entropy and entropy change due to the exchange of energy via heat.

Step-by-step explanation:

In an isothermal expansion of an ideal gas, energy must flow via heat into the gas from the surroundings to maintain the temperature. The total change in entropy is given bydSₜₒₜₐₗ = dSₛ_yₛₜₑₘ + dSₛᵤᵣᵣₒᵤₙ_dᵢₙ_gₛ. The entropy change for the system includes both the internally produced entropy and the entropy change due to the exchange of energy via heat. The heat exchanged adds to the total entropy because it represents an increase in disorder in the system.