Problem 44 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

\(\Delta_{t} G^{\circ}\) for the formation of \(\mathrm{HI}(\mathrm{g})\) from its gaseous elements is \(-2.303 \mathrm{kcal} / \mathrm{mol}\) at \(500 \mathrm{~K}\). When the partial pressure of HI is \(10 \mathrm{~atm}\) and of \(\mathrm{I}_{2}(\mathrm{~g})\) is \(0.001 \mathrm{~atm}\), what must be the partial pressure of hydrogen be at this temperature to reduce the magnitude of \(\Delta G\) for the reaction to zero? (a) \(1000 \mathrm{~atm}\) (b) \(10000 \mathrm{~atm}\) (c) \(100 \mathrm{~atm}\) (d) \(31.63 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The partial pressure of hydrogen must be 100 atm to reduce the magnitude of \(\Delta G\) for the reaction to zero.

Step 1: State the reaction and expression for the standard Gibbs free energy change

First, write down the balanced chemical equation for the formation of HI from its gaseous elements: \[\begin{equation} \frac{1}{2}H_{2(g)} + \frac{1}{2}I_{2(g)} \rightarrow HI_{(g)} \end{equation}\]The standard Gibbs free energy change \(\Delta_{t}G^\circ\) for this reaction is given as \(-2.303 \mathrm{kcal/mol}\).

Step 2: Write the expression for the Gibbs free energy change under non-standard conditions

The Gibbs free energy change under non-standard conditions, \(\Delta G\), can be calculated using the equation: \[\begin{equation} \Delta G = \Delta_{t}G^\circ + RT \ln Q \end{equation}\]where \(Q\) is the reaction quotient, \(R\) is the gas constant in appropriate units, and \(T\) is the temperature in kelvin.

Step 3: Calculate the reaction quotient (Q)

For the reaction given, the reaction quotient \(Q\) is given by: \[\begin{equation} Q = \frac{P_{\mathrm{HI}}}{\sqrt{P_{\mathrm{H2}} P_{\mathrm{I2}}}} \end{equation}\]Given, \(P_{\mathrm{HI}} = 10 \mathrm{~atm}\) and \(P_{\mathrm{I2}} = 0.001 \mathrm{~atm}\), substituting these into the reaction quotient gives us: \[\begin{equation} Q = \frac{10}{\sqrt{P_{\mathrm{H2}} \cdot 0.001}} \end{equation}\].

Step 4: Expression for \(\Delta G\) when its magnitude is reduced to zero

To reduce the magnitude of \(\Delta G\) to zero, we set \(\Delta G\) to zero and solve for \(P_{\mathrm{H2}}\): \[\begin{equation} 0 = -2.303 \mathrm{kcal/mol} + RT \ln Q \end{equation}\]Substitute the known values and solve for \(\ln Q\), then express \(Q\) in terms of \(P_{\mathrm{H2}}\) to find its value.

Step 5: Solve for the partial pressure of hydrogen (\(P_{\mathrm{H2}}\))

Rewrite the equation for \(\Delta G\) after converting \(\Delta_{t}G^\circ\) to joules and using the value of \(R = 8.314 \text{J/(mol K)}\): \[\begin{equation} 0 = -2404.18 \text{J/mol} + (8.314 \frac{\text{J}}{\text{mol K}})(500 \text{K}) \ln Q \end{equation}\]Solve for \(\ln Q\) and exponentiate to get \(Q\). Substitute \(Q\) back into the reaction quotient equation to find \(P_{\mathrm{H2}}\). The result is the pressure in atmospheres at which the reaction \(\Delta G\) would be zero.

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a state in which the concentrations of reactants and products remain constant over time, indicating a balance between forward and reverse reactions. This state occurs when the rate at which the reactants convert to products is equal to the rate at which the products revert back to reactants. At equilibrium, the amount of each substance does not have to be the same; it is the ratio of their concentrations that is fixed, described by the equilibrium constant, K.

Understanding the conditions of equilibrium can be pivotal when predicting the direction in which a reaction will proceed and its potential to make certain products in different environmental conditions. The equilibrium concept is directly tied to the Gibbs free energy, where a system at equilibrium has a Gibbs free energy change of zero. This ties into how reactions find a balance point where energy is neither being lost nor gained.

Reaction Quotient (Q)

The reaction quotient (Q) is a measure of the relative amounts of products and reactants present during a reaction at a particular point in time. It is used to predict the direction of a chemical reaction and to determine whether a system is at equilibrium. The formula for Q looks very much like the one for the equilibrium constant (K), but Q can be used at any point during a reaction, not just at equilibrium.

For a general reaction given by \( aA + bB \rightleftharpoons cC + dD \) where \( a \) , \( b \) , \( c \) , and \( d \) are the stoichiometric coefficients and \( A \) , \( B \) , \( C \) , and \( D \) are the reactants and products, respectively, the reaction quotient Q is defined as:
\[ Q = \frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}} \
\]\
where the concentrations of the gaseous components are often expressed in terms of partial pressures. If \( Q = K \) , the system is at equilibrium. If \( Q < K \) , the reaction will proceed in the forward direction to reach equilibrium, and if \( Q > K \) , the reaction will proceed in the reverse direction.

Standard Gibbs Free Energy Change

The standard Gibbs free energy change (\(\Delta G^\circ\)) is a thermodynamic property that indicates the amount of energy available to do work when a chemical reaction occurs at constant temperature and pressure unless otherwise specified. The 'standard' refers to reactions taking place under standard conditions (298 K and 1 bar pressure). It is calculated from the standard enthalpies and entropies of reactants and products.

The Gibbs free energy equation is given by:
\[ \Delta G = \Delta H - T\Delta S \
\]\
where \(\Delta H\) is the change in enthalpy, \(T\) is the temperature in Kelvin, and \(\Delta S\) is the change in entropy. When \(\Delta G^\circ\) is negative, the process is spontaneous in the forward direction under standard conditions; positive values indicate a non-spontaneous process, requiring energy to proceed.

In practical applications, when dealing with non-standard conditions, \( \Delta G \) is related to \( \Delta G^\circ \) and the reaction quotient (Q) via the equation:\[ \Delta G = \Delta G^\circ + RT \ln Q \
\]\
This relationship is crucial in predicting how changes in conditions, like temperature and concentration, affect the spontaneity of a given reaction.