Question

Consider the reaction $2{\mathrm{NO}}_2(g)⟶{\mathrm{N}}_2{\mathrm{O}}_4(g).$ (a) Using data from Appendix C, calculate $\mathrm{\Delta }{G}^{\circ }$ at $298\mathrm{K}$. (b) Calculate $\mathrm{\Delta }G$ at $298\mathrm{K}$ if the partial pressures of ${\mathrm{NO}}_2$ and ${\mathrm{N}}_2{\mathrm{O}}_4$ are $0.40\mathrm{atm}$ and $1.60\mathrm{atm}$, respectively.

Short answer

a) The standard Gibbs free energy change at 298 K is $\mathrm{\Delta }{G}^{\circ }=-4.7kJ/mol$. b) At the partial pressures of 0.40 atm for $N{O}_2$ and 1.60 atm for $N_2{O}_4$, the Gibbs free energy change is $\Delta G=-559J/mol$.

Step-by-step solution

Part (a): Calculate Standard Gibbs Free Energy Change at a Specific Temperature

1. Write the balanced chemical equation. $2N{O}_2(g)⟶N_2{O}_4(g)$ 2. Obtain standard Gibbs free energy of formation values for each substance. Look at Appendix C in the textbook or a reputable source (i.e. NIST) to find the standard Gibbs free energy of formation for each chemical at the given temperature (in this case, 298 K). $\mathrm{\Delta }{G}_f^{\circ }(N{O}_2)$ = $51.3kJ/mol$ $\mathrm{\Delta }{G}_f^{\circ }(N_2{O}_4)$ = $97.9kJ/mol$ 3. Calculate $\mathrm{\Delta }{G}^{\circ }$ for the reaction. Use the equation: $\mathrm{\Delta }{G}^{\circ }=\sum (n\mathrm{\Delta }{G}_f^{\circ }(products))-\sum (n\mathrm{\Delta }{G}_f^{\circ }(reactants))$ $\mathrm{\Delta }{G}^{\circ }=1\cdot 97.9-2\cdot 51.3=-4.7kJ/mol$

Part (b): Calculate ΔG at specific partial pressures

1. Use the expression for the reaction quotient Q. The reaction quotient Q = Qp, as both the reactant and product are in the gas phase and their pressures are given in the problem. Qp is defined as the ratio of the product of products' partial pressures, raised to their respective stoichiometric coefficients, to product of reactants' partial pressures, raised to their respective stoichiometric coefficients. $Q_p=\frac{(P_{N_2{O}_4}{)}^1}{(P_{N{O}_2}{)}^2}$ 2. Insert the given pressures and calculate Qp. $Q_p=\frac{(1.60\mathrm{atm}{)}^1}{(0.40\mathrm{atm}{)}^2}=10$ 3. Calculate ΔG at the given conditions. Use the equation: $\Delta G=\Delta G^{\circ }+RT\ln Q_p$ The standard free energy change $\Delta G^{\circ }$ and value of Qp are already known from steps earlier: $\Delta G^{\circ }$ = -4.7 kJ, Qp = 10. The universal gas constant R = 8.314 J/mol K, and the temperature T = 298 K. $\Delta G=-4700+(8.314\times 298)\ln 10=-4700+(\approx )4141=-559J/mol$ So, a) Standard Gibbs free energy change, $\mathrm{\Delta }{G}^{\circ }=-4.7kJ/mol$ b) Gibbs free energy change at given partial pressures, $\Delta G=-559J/mol$

Key concepts

Chemical Thermodynamics

Chemical thermodynamics deals with the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. It provides us with a framework for understanding how energy is transferred in the form of heat or work in chemical processes, which in turn dictates the direction and extent to which a chemical reaction can proceed.

At the heart of chemical thermodynamics is the concept of Gibbs Free Energy (denoted as G), which combines enthalpy, entropy, and temperature to provide a criterion for predicting the spontaneity of a process. A negative Gibbs Free Energy change $\mathrm{\Delta }G$ indicates a spontaneous process under constant pressure and temperature. To apply thermodynamics to a chemical reaction, the first step is usually to calculate the standard Gibbs Free Energy change $\mathrm{\Delta }{G}^{°}$ for the reaction, which tells us if the reaction would proceed spontaneously under standard conditions.

Reaction Quotient (Qp)

The reaction quotient, $Q_p$, plays a crucial role in predicting the direction of a chemical reaction at any given moment. It is particularly important when dealing with reactions involving gases, as it takes the form of partial pressures, hence denoted by the subscript p. The reaction quotient compares the current state of a system to the equilibrium state, represented by the equilibrium constant $K_p$.

When the values of $Q_p$ and $K_p$ are compared, one can ascertain which direction the reaction will shift to reach equilibrium. If $Q_p>K_p$ the reaction will proceed in the reverse direction to decrease the products, and if $Q_p<K_p$ it will move forward to produce more products. The calculation of $Q_p$ is similar to that of $K_p$ but uses non-equilibrium partial pressures of gas-phase reactants and products.

Standard Gibbs Free Energy of Formation

The standard Gibbs free energy of formation $\mathrm{\Delta }{G}_f^{°}$ is a fundamental quantity in chemical thermodynamics. It represents the change in Gibbs free energy when one mole of a compound is formed from its elements in their standard states. In essence, $\mathrm{\Delta }{G}_f^{°}$ is the thermodynamic potential used to calculate the maximum amount of non-expansion work that can be extracted from a chemical reaction at constant temperature and pressure.

When we analyze the feasibility of a chemical reaction, we consider the $\mathrm{\Delta }{G}_f^{°}$ values of its reactants and products. For a given reaction, the overall standard Gibbs free energy change $\mathrm{\Delta }{G}^{°}$ is calculated by subtracting the sum of the standard Gibbs free energies of the reactants from the sum of the standard Gibbs free energies of the products, taking into account their stoichiometric coefficients.

Equilibrium Partial Pressures

Equilibrium partial pressures are central to understanding the behavior of gas-phase reactions at equilibrium. At equilibrium, the rates of the forward and reverse reactions are equal, and the partial pressures of the reactants and products remain constant. The value of the equilibrium constant $K_p$ for a reaction is obtained by evaluating the equilibrium partial pressures of the reactants and products.

A vital aspect of the equilibrium condition is that while the partial pressures are constant, it does not imply all reactants and products have the same pressure, but their ratio at equilibrium equals the ratio defined by the equilibrium constant $K_p$. Determining the equilibrium partial pressures is critical when analyzing changes in conditions, such as changes in total pressure, temperature, or the addition of inert gases, all of which can shift the position of equilibrium.