How to Calculate Gibbs Free Energy Without Temperature

Calculating Gibbs free energy (G) without temperature is not possible since it is a thermodynamic property that depends on both enthalpy (H) and entropy (S) changes at a specific temperature (T). However, if you have the standard Gibbs free energy change (ΔG°) at a specific temperature and the equilibrium constant (K) for a reaction, you can estimate the Gibbs free energy at another temperature using the van’t Hoff equation.

The van’t Hoff Equation

The van’t Hoff equation relates the equilibrium constant (K) of a reaction to the standard enthalpy change (ΔH°) and temperature (T) using the gas constant (R):

ln (K2/K1) = (ΔH°/R) * (1/T1 - 1/T2)

Where:
– K1 and K2 are the equilibrium constants at temperatures T1 and T2, respectively
– ΔH° is the standard enthalpy change
– R is the gas constant (8.314 J/(mol·K))

By rearranging the equation, you can calculate the Gibbs free energy at temperature T2:

ΔG°(T2) = ΔG°(T1) + R * T2 * ln K2

This method allows you to estimate the Gibbs free energy at a different temperature if you know the standard Gibbs free energy change and the equilibrium constant at a specific temperature.

Theorem

The van’t Hoff equation relates the equilibrium constant (K) of a reaction to the standard enthalpy change (ΔH°) and temperature (T) using the gas constant (R).

Physics Formula

The van’t Hoff equation:

ln (K2/K1) = (ΔH°/R) * (1/T1 - 1/T2)

Physics Examples

Consider a reaction with a known standard Gibbs free energy change (ΔG°) of -50 kJ/mol at 298 K and an equilibrium constant (K) of 10^5. Using the van’t Hoff equation, you can estimate the Gibbs free energy at 310 K.

Physics Numerical Problems

Given a reaction with ΔH° = -100 kJ/mol, K1 = 10^3 at 300 K, and ΔG°(300 K) = -50 kJ/mol, calculate ΔG°(310 K).

To solve this problem, we can use the van’t Hoff equation:

ln (K2/K1) = (ΔH°/R) * (1/T1 - 1/T2)
ln (K2/10^3) = (-100 kJ/mol) / (8.314 J/(mol·K)) * (1/300 K - 1/310 K)
ln (K2/10^3) = -12.02
K2 = 10^3 * e^(-12.02) = 1.67 × 10^-6

Now, we can use the rearranged equation to calculate ΔG°(310 K):

ΔG°(310 K) = ΔG°(300 K) + R * T2 * ln K2
ΔG°(310 K) = -50 kJ/mol + (8.314 J/(mol·K)) * 310 K * ln (1.67 × 10^-6)
ΔG°(310 K) = -50 kJ/mol + 3.44 kJ/mol = -46.56 kJ/mol

Figures, Data Points, Values, Measurements

The van’t Hoff equation uses the following parameters to estimate the Gibbs free energy at a different temperature:

Standard enthalpy change (ΔH°)
Equilibrium constant (K) at a specific temperature (T1)
Desired temperature (T2)

By plugging in these values, you can calculate the equilibrium constant (K2) at the new temperature (T2) and then use it to find the Gibbs free energy (ΔG°) at that temperature.

References

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