Problem 22 - Chemistry: An Atoms-Focused Approach

Question

What happens to the internal energy of a gas when it expands (with no heat flow)?

Step-by-Step Solution

Answer

Answer: When a gas expands without any heat flow, its internal energy decreases. This is because the work done by the system during adiabatic expansion is at the expense of its internal energy.

Step 1: Recall the First Law of Thermodynamics

The first law of thermodynamics states that the change in internal energy (∆U) of a system is equal to the difference between the heat (Q) added to the system and the work (W) done by the system on its surroundings. Mathematically, we can represent it as: ∆U = Q - W Since there is no heat flow in this expansion (Q = 0), the equation becomes: ∆U = -W

Step 2: Relate Work Done to the Adiabatic Expansion

For an ideal gas undergoing an adiabatic expansion, the work done by the gas is given by the formula: W = (P1V1 - P2V2) / (γ - 1) where P1 and V1 are the initial pressure and volume, P2 and V2 are the final pressure and volume, and γ (gamma) is the adiabatic index or the ratio of specific heat capacities (Cp/Cv).

Step 3: Determine the Internal Energy Change

Now that we have related work to the adiabatic expansion, we can substitute the expression for work into the equation for internal energy change: ∆U = - (P1V1 - P2V2) / (γ - 1)

Step 4: Analyze the Internal Energy Change

When a gas expands adiabatically, its pressure decreases and its volume increases. Therefore, P2 < P1 and V2 > V1. From the equation in step 3, we can see that if the term (P1V1 - P2V2) is positive, the internal energy change (∆U) will be negative. Thus, during adiabatic expansion, the internal energy of the gas decreases and the work done by the system is at the expense of its internal energy. So, the internal energy of the gas decreases when it expands without any heat flow.