How do you calculate entropy?

Question

Avery Adams · Accepted Answer

These are a few formulas. #ΔS_"total" = ΔS_"univ" = ΔS_"sys" + ΔS_"surr" =  q_"sys"/T_"sys" + q_"surr"/T_"surr"# where #q# is the heat and #T# is the Kelvin temperature. Change in Entropy for the System #ΔS_"sys" = ΔS_"rxn" = Sigma(n_pS_"products"^"o") – Sigma(n_rS_"reactants"^"o")# where #n_p# and #n_r# represent moles of products and reactions.  Change in Entropy for the Environment #ΔS_"surr" = q_"surr"/T_"surr"# #q_"surr" = -q_"sys"# #ΔS_"surr" = q_"surr"/T_"surr" = -q_"sys"/T_"surr"# First example: What is #ΔS_"surr"#  at 300 K for the reaction reactants → products; #ΔH# = 75 kJ Resolution: #ΔS_"sys" = q/T = (75 000" J")/(300" K")# = 250 J/K #ΔS_"surr" = q_"surr"/T_"surr" =  -(ΔH_"sys")/T_"surr" = -(75 000" J")/(300" K")# = -250 J/K Second example: What is #ΔS_"rxn"^"o"# for the following reaction? 4NO(g) + 6H₂O(g) = 4NH₃(g) + 5O₂(g) The #S^"o"# values are NH₃ = 193 J·K⁻¹mol⁻¹; O₂ = 205 J·K⁻¹mol⁻¹; NO = 211 J·K⁻¹mol⁻¹;
H₂O = 189 J·K⁻¹mol⁻¹ #ΔS_"sys" = Sigma(n_pS_"products"^"o") – Sigma(n_rS_"reactants"^"o")# #ΔS_"rxn"^"o" = 4S_"NO"^"o" + 6S_"H₂O"^"o" – 4S_"NH₃" - 5S_"O₂"^"o"# #ΔS_"rxn"^"o" = (4×211 + 6 × 189 - 4 × 193 + 5 × 205 )# J/K⁻¹ = 181 J•K⁻¹

Caleb Adams · Answer

undefined Entropy can be calculated using the formula:

$$ S = -k \sum_{i} P_i \ln(P_i) $$

Where:
- $ S $ is the entropy
- $ k $ is the Boltzmann constant
- $ P_i $ is the probability of finding the system in the $ i^{th} $ microstate
- $ \ln $ denotes the natural logarithm

The formula sums over all microstates of the system, multiplying the probability of each microstate by the natural logarithm of that probability, and then takes the negative of the result multiplied by the Boltzmann constant.