Why change in internal energy is zero in isothermal process In isothermal process $\Delta U =0$. But I am having trouble understanding it.
Say we have an ideal gas, and say my temperature is constant but I move the pressure, volume from $(P, V) \to (P-dP, V+dV) $. So the volume has expanded and system has done some work to the surrounding. So my work is non-zero.
So how come $\Delta U=0$? I am really confused here.
 A: In Isothermal process the temperature is constant. 
The internal energy is a state function dependent on temperature. Hence, the internal energy change is zero.
For the process you are describing the work is done by the system, but had you not supplied heat, then the temperature would have dropped. That is a adiabtic cooling process. If no heat is supplied and internal energy is not maintained at the same level, then the process wont be a isothermal process.
A: Internal energy is due to motion of particles in a system. As internal energy depends on temperature. As we know temperature in isothermal process is constant so the internal energy will also be constant thus the change in internal energy will be zero. 
A: It is not generally true that $\Delta U = 0$ in an isothermal process.
An ideal gas by definition has no interactions between particles, no intermolecular forces, so pressre change at constant temperature does not change internal energy.
Real gases have intermolecular interactions, attractions between molecules at low pressure and repulsion at high pressure.  Their internal energy changes with change in pressure, even if temperature is constant.
For an ideal gas, in an isothermal process, $\Delta U = 0 = Q-W$, so $Q=W$.
A: You might have got the answer, but this answer is for new visitors ...
When you say isothermally it means the system is somehow allowed to exchange thermal or mechanical energy. In an isothermal compression, the system is allowed to release heat otherwise (adiabatic process) change in temperature will change the internal energy.
Similarly in isothermal expansion, the system does work on the expense of its internal energy which is compensated by influx of heat otherwise the temperature will decrease. 
For your question, the system must have absorbed some heat and the expansion work consumes it. This satisfies the equation: the inward heat is negative and the work is done by the system is positive both are equal so no net change in internal energy, yet the volume increases.
A: See consider a cylinder with piston fixed, (i.e. it doesn't move) and the system is provided with a source with temperature $T$. Now as the piston does not move the volume is constant, so no work is done and internal energy is also constant and no heat is added since system and source are at the same temperature.
Now let us release the piston. Then the system does work using internal energy, but the change in internal energy is spontaneously overcome since system is copped to maintain at constant temperature. In other words we can say that system never uses internal energy since it is supplied with constant heat from a source. 
