Question

Without referring to tables, predict which of the following has the higher enthalpy in each case: (a) \(1 \mathrm{~mol} \mathrm{CO}_{2}(s)\) or \(1 \mathrm{~mol}\) \(\mathrm{CO}_{2}(g)\) at the same temperature, (b) 2 mol of hydrogen atoms or \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2},\) (c) \(1 \mathrm{~mol} \mathrm{H}_{2}(g)\) and \(0.5 \mathrm{~mol} \mathrm{O}_{2}(g)\) at \(25^{\circ} \mathrm{C}\) or \(1 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}(g)\) at \(25^{\circ} \mathrm{C},\) (d) \(1 \mathrm{~mol} \mathrm{~N}_{2}(g)\) at \(100{ }^{\circ} \mathrm{C}\) or \(1 \mathrm{~mol}\) \(\mathrm{N}_{2}(g)\) at \(300^{\circ} \mathrm{C}\)

Short answer

(a) 1 mol of \(\mathrm{CO}_{2}(g)\) has a higher enthalpy. (b) 2 mol of hydrogen atoms have a higher enthalpy. (c) 1 mol of \(\mathrm{H}_{2}(g)\) and 0.5 mol of \(\mathrm{O}_{2}(g)\) have a higher combined enthalpy. (d) 1 mol of \(\mathrm{N}_{2}(g)\) at \(300^{\circ} \mathrm{C}\) has a higher enthalpy.

Step-by-step solution

Case (a): CO2(s) vs CO2(g)

As both the solid CO2 and gaseous CO2 are at the same temperature, the only difference between them is the phase. The enthalpy of the substance increases when it transitions from solid to liquid and then from liquid to gas. Therefore, 1 mol of \(\mathrm{CO}_{2}(g)\) will have a higher enthalpy than 1 mol of \(\mathrm{CO}_{2}(s)\) at the same temperature because it has absorbed more energy during the phase transition.

Case (b): 2 mol H atoms vs 1 mol H2

When 2 mol of hydrogen atoms combine to form 1 mol of \(\mathrm{H}_{2}\) molecule, a chemical bond is formed between the two hydrogen atoms. The energy is released during the formation of the bond, resulting in the decrease of enthalpy for the system. Hence, 2 mol of hydrogen atoms will have a higher enthalpy compared to 1 mol of \(\mathrm{H}_{2}\).

Case (c): 1 mol H2(g) + 0.5 mol O2(g) vs 1 mol H2O(g)

After the combustion reaction of hydrogen and oxygen, 1 mol of \(\mathrm{H}_{2}(g)\) and 0.5 mol of \(\mathrm{O}_{2}(g)\) would form 1 mol of \(\mathrm{H}_{2} \mathrm{O}(g)\). This reaction releases energy as enthalpy decreases: \[ \mathrm{H}_{2}(g) + \dfrac{1}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{H}_{2} \mathrm{O}(g)\] The enthalpy of 1 mol of \(\mathrm{H}_{2} \mathrm{O}(g)\) will be lower than the combined enthalpy of 1 mol of \(\mathrm{H}_{2}(g)\) and 0.5 mol of \(\mathrm{O}_{2}(g)\) at the same temperature, as energy has been released during the reaction.

Case (d): 1 mol N2(g) at 100 °C vs 1 mol N2(g) at 300 °C

As temperature increases, the enthalpy of the substance also increases because it has absorbed more energy to reach the higher temperature. The temperature is the only difference between these two cases. Therefore, 1 mol of \(\mathrm{N}_{2}(g)\) at \(300^{\circ} \mathrm{C}\) will have a higher enthalpy than 1 mol of \(\mathrm{N}_{2}(g)\) at \(100{ }^{\circ} \mathrm{C}\).

Key concepts

Phase Transition Enthalpy

When a substance undergoes a phase change, such as solid carbon dioxide (dry ice) sublimating into carbon dioxide gas (CO2), there is a change in its enthalpy. Enthalpy is essentially a measure of heat content, and when the phase transition occurs without a change in temperature, it implies that energy has been absorbed to overcome intermolecular forces.

During sublimation, solid CO2 requires energy to transition into the gaseous phase. In this endothermic process, the enthalpy increases because the molecules in the gas phase have more freedom to move and therefore have higher potential energy. This is why, at the same temperature, 1 mole of CO2 gas has a higher enthalpy than 1 mole of solid CO2.

Chemical Bond Energy

Chemical bond energy is the energy required to break a bond or the energy released when a bond is formed. When considering 2 moles of hydrogen atoms versus 1 mole of H2, we're looking at individual atoms versus bonded atoms. The diatomic molecule H2 is formed by a covalent bond between the two hydrogen atoms.

As this bond forms, energy is released, which is essentially the 'bond energy.' So, the system's enthalpy decreases. Consequently, before the formation of the H2 molecule, when the atoms are separate, the energy - and hence enthalpy - of the system is higher. This explains why 2 moles of hydrogen atoms have more enthalpy than 1 mole of H2.

Combustion Reaction Enthalpy

In a combustion reaction, substances react with oxygen to release heat, and typically, the process involves the formation of new chemical bonds. Taking the reaction where hydrogen gas combines with oxygen to form water vapor (H2O), the enthalpy change is significant.

Because the formation of water from hydrogen and oxygen releases energy - typically in the form of heat - the reaction is exothermic. This means the products, in this case, water vapor, will have a lower enthalpy than the reactants, hydrogen and oxygen gases. This is why after combustion, 1 mole of H2O(g) at 25°C has a lower enthalpy than the combined enthalpy of 1 mole of H2(g) and 0.5 mole of O2(g) at the same temperature.

Temperature Influence on Enthalpy

Temperature effectively measures the average kinetic energy of particles in a substance. With an increase in temperature, particles move faster and tend to occupy a larger volume. This kinetic activity corresponds to an increase in the system's enthalpy.

Thus, when comparing nitrogen gas (N2) at 100°C to nitrogen gas at 300°C, the molecules within the warmer sample are moving with greater energy, resulting in a higher enthalpy. Hence, 1 mole of N2(g) at 300°C has a higher enthalpy than 1 mole of N2(g) at 100°C, illustrating the direct relationship between temperature and enthalpy.