The following equation represents the decomposition of a generic diatomic element in its standard state.
(1/2)X2(g)--->X(g)
Assume that the standard molar Gibbs energy of formation of X(g) is 5.45 kJ·mol–1 at 2000. K and –64.30 kJ·mol–1 at 3000. K. Determine the value of K (the thermodynamic equilibrium constant) at each temperature.
at 2000K K=___
at 3000K K=___
Assuming that ΔH°rxn is independent of temperature, determine the value of ΔH°rxn from these data.
ΔH°rxn=___
I know i have to use:
∆G_rxn = - R∙T∙ln(K)
and,
K = e^{ - ∆G_r /(R∙T) } but I can't seem to get the correct answer am I missing somthing?
That looks ok to me.
Don't forget to convert from kJ to J or vice versa.
You are on the right track with using the equation ∆G_rxn = - R∙T∙ln(K) to determine the equilibrium constant at each temperature. However, there is a small mistake in the equation you presented for K. The correct equation is K = e^(-∆G_r /(R∙T)), where ∆G_r is the change in Gibbs energy for the reaction.
Now, let's calculate the equilibrium constant (K) at each temperature using the given values.
1. At 2000K:
∆G_r = 5.45 kJ·mol^(-1) (from the given standard molar Gibbs energy of formation of X(g) at 2000K)
Using the equation ∆G_rxn = - R∙T∙ln(K), we can rearrange it to solve for K:
K = e^(-∆G_r /(R∙T))
where R is the gas constant (8.314 J·mol^(-1)·K^(-1)).
Plugging in the values:
K(2000K) = e^(-5.45 kJ·mol^(-1) / (8.314 J·mol^(-1)·K^(-1) * 2000 K))
Note: The units of R and ∆G_r must be consistent, so we converted kJ to J and KJ to J in the equation.
Now, let's calculate it:
K(2000K) = e^(-5450 J·mol^(-1) / (8.314 J·mol^(-1)·K^(-1) * 2000 K))
K(2000K) ≈ 3.71
Therefore, the equilibrium constant (K) at 2000K is approximately 3.71.
2. At 3000K:
∆G_r = -64.30 kJ·mol^(-1) (from the given standard molar Gibbs energy of formation of X(g) at 3000K)
Using the same equation, we can calculate K(3000K):
K(3000K) = e^(-∆G_r /(R∙T))
K(3000K) = e^(-(-64.30 kJ·mol^(-1)) / (8.314 J·mol^(-1)·K^(-1) * 3000 K))
K(3000K) ≈ 1.70 × 10^18
Therefore, the equilibrium constant (K) at 3000K is approximately 1.70 × 10^18.
Now, to determine ΔH°rxn, we can use the equation:
ΔG°rxn = -RTln(K)
Since ΔG°rxn = ΔH°rxn - TΔS°rxn, and assuming ΔH°rxn is independent of temperature, ΔS°rxn can be neglected. Therefore, we have:
ΔH°rxn = ΔG°rxn
Now, let's calculate ΔH°rxn:
ΔH°rxn = ΔG°rxn = - R * T * ln(K)
Using the given values at 2000K:
ΔH°rxn = - (8.314 J·mol^(-1)·K^(-1)) * (2000 K) * ln(3.71)
ΔH°rxn ≈ - 31.4 kJ·mol^(-1)
Therefore, ΔH°rxn is approximately -31.4 kJ·mol^(-1) based on the given data.
To find the value of K (the thermodynamic equilibrium constant) at each temperature, we need to use the equation ΔG_rxn = -R*T*ln(K), where ΔG_rxn is the change in Gibbs energy, R is the gas constant (8.314 J·mol^−1·K^−1), T is the temperature in Kelvin, and K is the equilibrium constant.
Given that the standard molar Gibbs energy of formation of X(g) is 5.45 kJ·mol^−1 at 2000 K and -64.30 kJ·mol^−1 at 3000 K, we need to convert these values to Joules:
ΔG_rxn at 2000 K = 5.45 kJ·mol^−1 = 5.45 * 10^3 J·mol^−1
ΔG_rxn at 3000 K = -64.30 kJ·mol^−1 = -64.30 * 10^3 J·mol^−1
Now we can calculate the value of K at each temperature using the equation K = e^(-ΔG_rxn / (R*T)):
At 2000 K:
K = e^(-5.45 * 10^3 J·mol^−1 / (8.314 J·mol^−1·K^−1 * 2000 K))
= e^(-2.60)
At 3000 K:
K = e^(-(-64.30 * 10^3 J·mol^−1) / (8.314 J·mol^−1·K^−1 * 3000 K))
= e^(21.71)
To calculate the value of ΔH_rxn (the change in enthalpy), we can use the equation ΔH_rxn = ΔG_rxn + TΔS_rxn, where ΔG_rxn is the change in Gibbs energy, T is the temperature in Kelvin, and ΔS_rxn is the change in entropy.
Since the problem states that ΔH_rxn is independent of temperature, we can assume that ΔS_rxn is also independent of temperature. Therefore, at any given temperature, ΔH_rxn = ΔG_rxn.
Therefore, ΔH_rxn = -5.45 kJ·mol^−1 at 2000 K and ΔH_rxn = -64.30 kJ·mol^−1 at 3000 K.
So, the value of ΔH_rxn is -5.45 kJ·mol^−1 at 2000 K and -64.30 kJ·mol^−1 at 3000 K.