Question

Given that \(\mathrm{H}(g)+\mathrm{H}(g) \longrightarrow \mathrm{H}_{2}(g) \quad \Delta H^{\circ}=-436.4 \mathrm{~kJ}\) calculate the change in mass (in kg) per mole of \(\mathrm{H}_{2}\) formed.

Short answer

After calculate from the equation in Step 3, the change of mass per mole of \(\mathrm{H}_2\) formed is approximately \(-4.86 \times 10^{-12}\) kg. The negative sign indicates that mass is lost in the process.

Step-by-step solution

Convert the Entropy Change into Joules

The provided data is in kilojoules (kJ). Since the equation requires the data in joules (J), convert kilojoules to joules: \(\Delta H^{\circ}= -436.4 \mathrm{~kJ}\) or \(\Delta H^{\circ} = -436.4 \times 1,000 \mathrm{~J} = -436,400 \mathrm{~J}\). The negative sign indicates that energy is lost during the reaction.

Use Einstein's Mass-Energy Equivalence Equation

Apply Einstein's mass-energy equivalence equation (\(E=mc^2\)) to convert energy loss to mass equivalence. In this case, the energy E is the energy change \(\Delta H^{\circ}\) and c is the speed of light which is approximately \(3.00 \times 10^8 \mathrm{~m/s}\). We want to solve for m (mass equivalent of the energy change). Therefore, \(m = \frac{E}{c^2}\). Substituting for \(E = \Delta H^{\circ} = -436,400 \mathrm{~J}\) and \(c = 3.00 \times 10^8 \mathrm{~m/s}\), the formula becomes \(m = \frac{-436,400}{(3.00 \times 10^8)^2}\).

Calculate for Mass Change

Calculate for m using the values from step 2. The resulting m is the change in mass per mole of \(\mathrm{H}_2\) formed and it‘s expressed in kilogram (kg).

Key concepts

Enthalpy Change

Understanding enthalpy change is fundamental in chemical thermodynamics. It represents the heat released or absorbed at a constant pressure during a chemical reaction. It's symbolized by \(\Delta H\) and is measured in joules per mole (J/mol). For exothermic reactions, where energy is released to the surroundings, \(\Delta H\) is negative, just like in the reaction given where hydrogen gas combines to form molecular hydrogen, \(\mathrm{H}_2\).

When hydrogen atoms bond to form \(\mathrm{H}_2\), they release energy, and \(\Delta H\) quantifies this energy change. In the textbook exercise, the change is \( -436.4 \) kJ per mole of \(\mathrm{H}_2\) formed, signifying an exothermic process. This knowledge is essential when calculating the change in mass that accompanies the reaction, as per Einstein's mass-energy equivalence principle.

Einstein's Mass-Energy Equivalence

Einstein's groundbreaking formula, \(E = mc^2\), where \(E\) represents energy, \(m\) stands for mass, and \(c\) is the speed of light, reveals the intimate relationship between mass and energy: they are convertible. This principle suggests that even a tiny amount of mass can be converted into a significant amount of energy, and vice versa.

This law of physics is pivotal in explaining how, during a chemical reaction like the formation of molecular hydrogen, there’s a tiny, but measurable, mass loss as energy is released. It's a cornerstone concept when it comes to comprehending the energy to mass conversion in reactions, and it guides us in calculating the change in mass in our given exercise.

Stoichiometry of Chemical Reactions

Stoichiometry is all about the quantitative relationships between the reactants and products in a chemical reaction. It helps chemists determine the amount of substance needed or produced in a reaction. The concept applies various laws including the conservation of mass and the fixed proportions of elements in compounds to deduce these relationships.

In the case of the hydrogen reaction, stoichiometry dictates that two moles of hydrogen gas combine in a fixed proportion to form one mole of molecular hydrogen. Understanding stoichiometry is not only key in predicting how much product will form from given amounts of reactants but also essential in interpreting the enthalpy change per mole, which in turn is needed to calculate the corresponding mass change.

Energy to Mass Conversion

In chemical processes, while most of the time the change in mass due to energy changes is exceptionally small, it is nonetheless a real phenomenon. The exercise showcases a direct application of this concept using the energy released from a chemical reaction—the formation of molecular hydrogen—to calculate the mass equivalent of that energy.

This energy to mass conversion, grounded in \(E=mc^2\text{)}\), may not cause detectable weight changes on a typical lab scale. However, it's crucial for understanding nuclear reactions, like those in the sun, where the conversion of mass to energy as per Einstein's formula powers the star's luminosity. In our exercise, applying this principle allows us to find out the minuscule change in mass associated with the formation of one mole of hydrogen gas.