Calculation of First-Law Quantities II

For an ideal gas, both internal energy and enthalpy are state functions that depend solely on temperature. This means any change in temperature directly results in changes to U and H. Since state functions depend only on initial and final states, not the path taken, calculating ΔU and ΔH requires only temperature values, making these calculations straightforward regardless of process type.

The change in internal energy is calculated by integrating the heat capacity at constant volume over the temperature range: ΔU = ∫Cv(T)dT from initial to final temperature. This integration method applies to any process type—reversible or irreversible—because internal energy is a state function depending only on temperature change, not the specific path taken.

State functions like internal energy and enthalpy depend only on initial and final states, so their changes are independent of the path taken. Path functions like heat and work depend on the specific process followed between states. For example, q and w vary with different reversible or irreversible paths, but ΔU and ΔH remain constant for the same temperature change.

For a reversible process, work is calculated as w = −∫pdV. For an ideal gas, pressure is replaced with nRT/V, allowing the integral to be evaluated over the volume range. Once work is determined, the first law of thermodynamics allows you to calculate the heat transfer q from the relationship ΔU = q + w.

In a reversible isothermal process, temperature remains constant. Since internal energy and enthalpy depend only on temperature, both ΔU and ΔH equal zero. Consequently, from the first law, q = −w, meaning all heat absorbed by the system equals the work done by the system.

A reversible adiabatic process is defined as one where no heat is exchanged with the surroundings, so q = 0. Under this condition, the first law simplifies to ΔU = w, meaning all internal energy change results from work done on or by the system. Energy changes are calculated by integrating heat capacity over the temperature range.

The first law states that ΔU = q + w, where ΔU is the change in internal energy, q is heat exchanged, and w is work performed. This equation shows that internal energy change equals the sum of heat and work. For ideal gases, once you calculate ΔU from temperature change and determine w from the process path, you can find q using this fundamental relationship.