$$4.0$$ L of an ideal gas is allowed to expand isothermally into vacuum until the total volume is $$20$$ L. The amount of heat absorbed in this expansion is ______ L atm.

We are asked to find the heat absorbed when 4.0 L of an ideal gas expands isothermally into vacuum until the total volume reaches 20 L. Because the gas expands into a vacuum (free expansion), the external pressure is zero ($$P_{ext} = 0$$). Work done during the expansion against external pressure is given by

$$w = -P_{ext} \times \Delta V$$

Substituting $$P_{ext} = 0$$ and $$\Delta V = 20 - 4$$ yields

$$w = -0 \times (20 - 4) = 0$$

so no work is performed during free expansion.

The first law of thermodynamics states that

$$\Delta U = q + w$$

For an ideal gas undergoing an isothermal (constant temperature) process, the internal energy depends only on temperature, so $$\Delta T = 0$$ and therefore $$\Delta U = 0$$.

Substituting $$\Delta U = 0$$ and $$w = 0$$ into the first law gives

$$0 = q + 0$$

which implies

$$q = 0$$

Thus, in a free expansion of an ideal gas under isothermal conditions, both work and heat exchange are zero. The correct answer is 0 L atm.