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Internal energy is a state function dependent on temperature, thus, when temperature does not change (definition of isothermal), there is no change in internal energy. Additional note: Dealing with ideal gases, contributions to U are kinetic energy and intermolecular interactions. The latter is 0, because we are dealing with ideal gases, and so we are left with kinetic energy. The kinetic energy does not change if the temperature does not change.
In the case of an ideal gas, the change in internal energy is given by delta U = (3/2)*R*delta T. If a reaction involving an ideal gas is isothermal, the temperature remains constant, meaning delta T=0. Hence, plugging delta T=0 in the aforementioned delta U equation gives delta U=0.
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