Question

which of the following processes always results in an increase in the energy of a system?
o the system loses heat and does work on the surroundings.
o the system gains heat and does work on the surroundings.
o the system loses heat and has work done on it by the surroundings.
o the system gains heat and has work done on it by the surroundings.
o none of these is always true.

Explanation:

Step1: Analyze the first law of thermodynamics

The first law of thermodynamics is $\Delta U = Q+W$, where $\Delta U$ is the change in internal energy of the system, $Q$ is the heat added to the system (positive if heat is gained by the system, negative if heat is lost by the system), and $W$ is the work done on the system (positive if work is done on the system, negative if work is done by the system).

Step2: Analyze each option

• For the option “The system gains heat and has work done on it by the surroundings”: If $Q>0$ (heat gain) and $W>0$ (work done on the system), then $\Delta U=Q + W>0$, so the internal energy of the system increases.
• For the option “The system loses heat and has work done on it by the surroundings”: If $Q<0$ (heat loss) and $W>0$ (work done on the system), $\Delta U=Q + W$. The sign of $\Delta U$ depends on the magnitudes of $Q$ and $W$. For example, if $|Q|>W$, $\Delta U<0$; if $|Q|0$.
• For the option “The system gains heat and does work on the surroundings”: If $Q>0$ (heat gain) and $W<0$ (work done by the system), $\Delta U=Q+W$. The sign of $\Delta U$ depends on the magnitudes of $Q$ and $W$. For example, if $Q <|W|$, $\Delta U<0$; if $Q>|W|$, $\Delta U>0$.
• For the option “The system loses heat and does work on the surroundings”: If $Q<0$ (heat loss) and $W<0$ (work done by the system), then $\Delta U=Q + W<0$, so the internal energy of the system decreases.

Answer:

The system gains heat and has work done on it by the surroundings.