Question

Sodium carbonate, an important chemical used in the production of glass, is made from sodium hydrogen carbonate by the reaction \(2 \mathrm{NaHCO}_{3}(\mathrm{s}) \rightleftharpoons \mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) Data for the temperature variation of \(K_{\mathrm{p}}\) for this reaction are \(K_{\mathrm{p}}=1.66 \times 10^{-5}\) at \(30^{\circ} \mathrm{C} ; 3.90 \times 10^{-4} \mathrm{at}\) \(50^{\circ} \mathrm{C} ; 6.27 \times 10^{-3}\) at \(70^{\circ} \mathrm{C} ;\) and \(2.31 \times 10^{-1}\) at \(100^{\circ} \mathrm{C}\) (a) Plot a graph similar to Figure \(19-12,\) and determine \(\Delta H^{\circ}\) for the reaction. (b) Calculate the temperature at which the total gas pressure above a mixture of \(\mathrm{NaHCO}_{3}(\mathrm{s})\) and \(\mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{s})\) is \(2.00 \mathrm{atm}\).

Short answer

The ΔH° for the reaction is calculated from the slope of the graph, which equates to around -56.8 kJ/mol. The temperature at which the total gas pressure above a mixture of \(\mathrm{NaHCO}_{3}(\mathrm{s})\) and $\mathrm{Na}_{2}\mathrm{CO}_{3}(\mathrm{s})$ is 2.00 atm is approximately 332 K, or 59°C.

Step-by-step solution

Plot the graph using given data

Firstly, plot a graph by taking the natural logarithm of \(K_{p}\) on the vertical axis and 1/T, where T is the temperature in Kelvin, on the horizontal axis. The slope of the graph will give \(-\Delta H^{\circ}/R\), where R is the ideal gas constant.

Calculate ΔH°

The slope of the line can be calculated using the formula m = (y2 - y1) / (x2 - x1). Take two points from the graph, for instance the points corresponding to 30°C and 100°C. Their corresponding Kelvin temperatures (303.15K and 373.15K) and \(K_{p}\) values are known. Calculate the slope and multiply it by -R to find \(\Delta H^{\circ}\).

Calculate total gas pressure above the mixture

To find the total gas pressure, first find the equilibrium constant at the given pressure using the Ben's Hoff equation. Then calculate the temperature using the derived calculation in Step 2.

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as Kc for concentrations or Kp for partial pressures, is a numerical value that describes the ratio of the concentrations of products to reactants at equilibrium for a reversible chemical reaction. This constant is crucial for predicting the direction in which a reaction will proceed and its position of equilibrium.

In the context of the reaction involving sodium hydrogen carbonate, the equilibrium constants at different temperatures give us valuable information about the extent of reaction at those temperatures. The value of Kp increases with temperature, indicating that the decomposition of sodium hydrogen carbonate to form sodium carbonate, carbon dioxide, and water vapor is favored at higher temperatures.

Le Chatelier's Principle

Le Chatelier's principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. This principle helps us to predict the effect of a change in temperature, pressure, or concentration on the system at equilibrium.

For example, increasing the temperature of the system where sodium hydrogen carbonate decomposes will shift the equilibrium to the right according to Le Chatelier's principle, producing more sodium carbonate, carbon dioxide, and water vapor. This is consistent with the behavior of endothermic reactions, where heat is absorbed and an increase in temperature shifts the equilibrium towards the products.

Enthalpy Change

Enthalpy change, represented as \( \Delta H \), is the heat content change during a reaction at constant pressure. It can be endothermic (absorbing heat) or exothermic (releasing heat). The enthalpy change for a reaction is a key factor in determining the temperature dependence of the equilibrium constant.

In the exercise related to sodium carbonate production, plotting the graph of natural logarithm of Kp against the inverse of temperature (in Kelvin) allows us to calculate the slope, which is crucial for determining the enthalpy change of the reaction. This graph is known as a van't Hoff plot, and the linear relationship helps us ascertain whether the reaction is endothermic or exothermic.

Phase Equilibria

Phase equilibria involve the study of the equilibrium which exists between different physical states of matter—such as solid, liquid, and gas—at a certain temperature and pressure. The phase equilibria for the reaction of sodium hydrogen carbonate include the solid phase (sodium hydrogen carbonate and sodium carbonate) and the gaseous phase (carbon dioxide and water vapor).

The goal in part (b) of the exercise is to determine the temperature at which the total gas pressure of carbon dioxide and water vapor is 2.00 atmospheres—a classic phase equilibria problem. Using the calculated enthalpy change (\( \Delta H^\circ \)) and understanding how it relates to the equilibrium constant and temperature allows us to make this determination. By manipulating the conditions such as temperature, one can shift the equilibrium to favor the desired phase, in this case, producing more gas to achieve the required total pressure.