Entropy calculations

Entropy as a state-function calculation

Entropy change is defined for a reversible heat transfer by

Delta S=int_i^f rac{delta Q_{rev}}{T}

Because entropy is a state function, this formula can be used with any convenient reversible path between the initial and final states. The actual process may be irreversible, but the entropy change depends only on endpoints.

This makes entropy calculations both powerful and subtle.

Constant-temperature heat transfer

If heat $Q_{rev}$ is transferred reversibly at constant temperature $T$, then

Delta S=rac{Q_{rev}}{T}

This is common during ideal phase changes, such as melting or boiling at constant temperature.

For melting,

Delta S=rac{mL_f}{T_m}

For vaporization,

Delta S=rac{mL_v}{T_b}

where temperatures must be in kelvins.

Heating a substance

If a substance with constant heat capacity $C$ is heated reversibly from $T_i$ to $T_f$, then

$delta Q_{rev}=C,dT$

so

ight)$$ For mass $m$ and specific heat $c$, $$Delta S=mclnleft(rac{T_f}{T_i} ight)$$ This result shows that entropy change depends on the ratio of temperatures, not just the difference. ## Ideal gas entropy change For an ideal gas, a useful formula is $$Delta S=nC_Vlnleft(rac{T_f}{T_i} ight)+nRlnleft(rac{V_f}{V_i} ight)$$ Another equivalent form is $$Delta S=nC_Plnleft(rac{T_f}{T_i} ight)-nRlnleft(rac{P_f}{P_i} ight)$$ These formulas are useful for ideal gas processes between equilibrium states. ## Isothermal ideal gas expansion For an isothermal ideal gas process, $T_f=T_i$, so the temperature term vanishes. The entropy change is $$Delta S=nRlnleft(rac{V_f}{V_i} ight)$$ Expansion increases entropy because the gas molecules have more accessible positions. Compression decreases the gas entropy, but the surroundings must compensate if the process is physically allowed. ## Reservoir entropy A thermal reservoir is so large that its temperature remains effectively constant when heat is exchanged. If a reservoir absorbs heat $Q$, its entropy change is $$Delta S_{res}=rac{Q}{T}$$ If it loses heat, $Q$ is negative. Reservoir entropy calculations are essential for finding entropy change of the universe. ## Mixing and irreversibility Entropy increases when gases mix because the number of accessible molecular arrangements increases. For ideal gases, mixing entropy can be calculated from changes in accessible volume for each gas. The key physical idea is that after mixing, molecules have more spatial possibilities. Mixing is spontaneous because the mixed macrostate corresponds to vastly more microstates than the separated macrostate. ## Units and signs Entropy has SI units of joules per kelvin: $$J/K$$ Positive entropy change means the system's entropy increases. Negative entropy change means it decreases. The second law requires the total entropy change of system plus surroundings to be nonnegative for real processes. ## The big idea Entropy calculations use reversible paths and absolute temperature. Common cases include constant-temperature phase changes, heating with heat capacity, ideal gas expansions, and reservoir exchanges. These calculations turn the second law into a quantitative tool for analyzing real thermal processes.