Problem 69 - Chemistry The Central Science

Question

Consider the following reaction between oxides of nitrogen: $$ \mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 3 \mathrm{NO}(g) $$ (a) Use data in Appendix C to predict how \(\Delta G\) for the reaction varies with increasing temperature. (b) Calculate \(\Delta G\) at \(800 \mathrm{K},\) assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not change with temperature. Under standard conditions is the reaction spontaneous at 800 \(\mathrm{K} ?\) (c) Calculate \(\Delta G\) at 1000 \(\mathrm{K} .\) Is the reaction spontaneous under standard conditions at this temperature?

Step-by-Step Solution

Answer

(a) From Appendix C, the signs of ΔH and ΔS for the given reaction are positive. Therefore, as temperature increases, ΔG will decrease. (b) For the reaction at 800 K, using the Van't Hoff equation with provided ΔH and ΔS values, calculate ΔG and find that it is positive, so the reaction is non-spontaneous at 800 K. (c) For the reaction at 1000 K, using the Van't Hoff equation, calculate ΔG and find that it is negative. Thus, the reaction is spontaneous under standard conditions at 1000 K.

Step 1: Determine the sign of ΔH and ΔS for the reaction

To predict the effect of temperature on ΔG, we need to determine the signs of ΔH and ΔS for the given reaction using the provided data in Appendix C. Note the value of ΔH and ΔS for each species involved in the reaction and calculate the overall ΔH and ΔS for the reaction.

Step 2: Analyze how increasing temperature will affect ΔG

Depending on the signs of ΔH and ΔS, we can predict how ΔG will vary with increasing temperature. For instance, if ΔH is positive and ΔS is positive, ΔG will decrease with increasing temperature. If ΔH is negative and ΔS is negative, ΔG will increase with increasing temperature. (b) Calculation of ΔG at 800 K To calculate ΔG at 800 K, we will use the Van't Hoff equation and the determined values of ΔH and ΔS:

Step 1: Find the values of ΔH and ΔS at 800 K

Assuming that ΔH and ΔS do not change with temperature, use the determined values from part (a) Step 1 for the given reaction.

Step 2: Calculate ΔG at 800 K

Plug in the values of ΔH, ΔS, and T (800 K) into the Van't Hoff equation to calculate ΔG at 800 K.

Step 3: Determine reaction spontaneity at 800 K

If ΔG is negative under standard conditions at 800 K, the reaction is spontaneous. If ΔG is positive, the reaction is non-spontaneous. (c) Calculation of ΔG at 1000 K

Step 1: Calculate ΔG at 1000 K

Plug in the values of ΔH, ΔS, and T (1000 K) into the Van't Hoff equation to calculate ΔG at 1000 K.

Step 2: Determine reaction spontaneity at 1000 K

Determine if the reaction is spontaneous or non-spontaneous under standard conditions at 1000 K based on the calculated value of ΔG at this temperature.

Key Concepts

Van't Hoff Equation

When studying chemical reactions, the Van't Hoff equation is crucial for understanding how changes in temperature affect the Gibbs free energy (ΔG). This equation helps us predict whether a reaction will be spontaneous at different temperatures.

The Van't Hoff equation is represented by: \[ \text{ΔG} = \text{ΔH} - T\text{ΔS} \]

Application to Calculate ΔG

To calculate Gibbs free energy changes, ΔH (enthalpy change) and ΔS (entropy change) of the reaction must be known. The temperature (T) at which the reaction occurs must also be specified. This information allows us to compute ΔG and determine the spontaneity of the reaction. Importantly, if ΔG is found to be negative, the process is spontaneous; if positive, the process is non-spontaneous.

The equation is particularly useful because it gives insight into how sensitive a reaction is to temperature changes. Higher temperatures generally favor reactions where ΔS is positive, due to the TΔS term increasing and thus potentially making ΔG more negative, indicating spontaneity. Conversely, if ΔS is negative, increasing the temperature may make ΔG less negative or even positive, suggesting non-spontaneity.

Gibbs Free Energy (ΔG)

Gibbs free energy (ΔG) serves as a predictive tool for the spontaneity of a chemical reaction under constant pressure and temperature. It incorporates both entropy (ΔS) and enthalpy (ΔH), giving a holistic view of a reaction's tendency to occur spontaneously.

ΔG provides important information:

• A negative value indicates the reaction is spontaneous.
• A positive value indicates the reaction is non-spontaneous.
• A value of zero implies the system is at equilibrium.

Spontaneity and ΔG

Spontaneity does not necessarily mean speed; it means the thermodynamic favorability of a process. For the reaction between oxides of nitrogen, determining ΔG using the provided enthalpy and entropy values from Appendix C will reveal whether the reaction tends to occur by itself at a given temperature. Thus, understanding ΔG is essential for predicting reaction behavior.

Reaction Temperature Effects

Temperature unquestionably plays a pivotal role in chemical reactions, affecting the rate at which reactions occur as well as their spontaneity. The Van't Hoff equation shows that both ΔH and ΔS can influence how a reaction's Gibbs free energy (ΔG) varies with temperature.

For example, if both ΔH and ΔS are positive, an increase in temperature will result in a larger TΔS term, potentially decreasing ΔG and making a reaction that was non-spontaneous at lower temperatures spontaneous at higher temperatures. In contrast, if ΔH is negative and ΔS is negative, increasing temperature can increase ΔG, making the reaction less likely to be spontaneous as temperature rises. Thus, carefully analyzing temperature effects is vital for making accurate predictions about reaction spontaneity under various conditions.

Thermodynamic Spontaneity

Thermodynamic spontaneity indicates whether a reaction can occur without an external input of energy. This concept is integral to understanding chemical equilibria and reaction directions. Spontaneity is influenced by enthalpy, entropy, and temperature—conceptually represented by Gibbs free energy (ΔG).

However, a spontaneous reaction isn't necessarily a fast one. This common misconception can mislead students when they assess chemical reaction properties. Thermodynamic spontaneity only asserts the potential of a reaction to occur, but not the rate at which it will proceed. The rate depends on kinetic factors like activation energy and the presence of a catalyst.

By examining the sign and magnitude of ΔG, we gain insights into the balance of energy and disorder within a system. This balance helps predict whether the system will change in order to reach a lower energy state or a state of higher entropy, thus indicating the potential for a spontaneous reaction.