# Can someone clear this confusion I have about the first law of thermodynamics

In our school we were taught that the first law of thermodynamics represented mathematically is $$\Delta U=Q+W$$. Also we were taught that for an isothermal process $$\Delta U=0$$ and therefore $$Q=-W$$, but we were taught that for an isothermal free expansion of a gas against vacuum $$W=0$$ and therefore $$\Delta U=Q$$ but shouldn't $$\Delta U$$ be zero here as well, I mean after all the process is still isothermal. Can somebody pls help with regards to this, also pls try to explain in a simple manner (High School Level). Thanks for the help

Everything you've written is correct (for a free expansion, $$\Delta U = q = w = 0$$), but it applies only if the system is an ideal gas, or has ideal-gas-like properties, namely that there are no attractive or repulsive interactions between the particles.
Thus, if you expand an ideal gas isothermally into a vacuum, the system doesn't do any work, so $$w=0$$. And there is thus no need to flow heat into the system to compensate for the work to maintain the temperature, nor is there any need to flow heat into the system to provide energy to pull the particles apart (no interaction between the particles), so $$q=0$$.