Question

Hydrogen gives off \(120 . \mathrm{J} / \mathrm{g}\) of energy when burned in oxygen, and methane gives off \(50 . \mathrm{J} / \mathrm{g}\) under the same circumstances. If a mixture of \(5.0 \mathrm{~g}\) hydrogen and \(10 . \mathrm{g}\) methane is burned, and the heat released is transferred to \(50.0 \mathrm{~g}\) water at \(25.0^{\circ} \mathrm{C}\), what final temperature will be reached by the water?

Short answer

The final temperature of the water can be found by following the steps provided in the solution. First, calculate the total heat released from the hydrogen and methane mixture: Q = Q_h + Q_m = \(5.0g × 120 \frac{J}{g} + 10.0g × 50 \frac{J}{g}\) = 600 J + 500 J = 1100 J Next, calculate the temperature change in the water using the equation: ΔT = \(\frac{Q}{m_{water} × C_{water}}\) = \(\frac{1100J}{50.0g × 4.18 \frac{J}{g °C}}\) = 5.26°C Finally, find the final temperature of the water: T_final = T_initial + ΔT = 25.0°C + 5.26°C = 30.26°C

Step-by-step solution

Calculate the total heat released from the hydrogen and methane mixture

To do this, we'll multiply the mass of each gas by its respective energy release per gram. Total heat released (Q) = Heat released from hydrogen (Q_h) + Heat released from methane (Q_m) First, we calculate the heat released by the hydrogen: Q_h = mass of hydrogen × energy released per gram of hydrogen Q_h = \(5.0g × 120 \frac{J}{g}\) Then, we calculate the heat released by the methane: Q_m = mass of methane × energy released per gram of methane Q_m = \(10.0g × 50 \frac{J}{g}\) Finally, we add the heats released by hydrogen and methane to get the total heat released.

Calculate the total heat released from the mixture

Now that we know the heat released by hydrogen and methane, we can find the total heat released by the mixture: Q = Q_h + Q_m = \(5.0g × 120 \frac{J}{g} + 10.0g × 50 \frac{J}{g}\)

Calculate the temperature change in the water

To find the temperature change in the water, we'll use the equation: Q = m_water × C_water × ΔT where Q is the heat absorbed by the water, m_water is the mass of the water, C_water is the specific heat capacity of water (4.18 J/g°C), and ΔT is the temperature change in the water. We can rearrange this equation to solve for ΔT: ΔT = \(\frac{Q}{m_{water} × C_{water}}\)

Calculate the final temperature of the water

Now, we can find the final temperature of the water (T_final) by adding the temperature change (ΔT) to the initial temperature of the water (T_initial): T_final = T_initial + ΔT = 25.0°C + ΔT Plug in the values for the temperature change that we calculated in step 3.

Key concepts

Enthalpy

In thermochemistry, enthalpy is a central concept that represents the total heat content of a system. It is commonly denoted by the letter "H" and is a state function, which depends only on the initial and final states of a system, not on the path taken to get there.
Understanding enthalpy is crucial because it helps us assess the heat transfer during chemical reactions.
For instance, in the combustion of hydrogen and methane, the enthalpy change gives us insights into how much heat is released:

• The enthalpy change (\( \Delta H \)) for exothermic reactions is negative since heat is released.
• Conversely, in endothermic reactions, \( \Delta H \) is positive as the system absorbs heat.

In our exercise, hydrogen and methane each have specific enthalpy changes, indicative of the energy they release upon combustion. By calculating the heat for individual components, one can determine the total enthalpy change of the system.

Specific Heat Capacity

Specific heat capacity is another vital concept in understanding how substances react to thermal changes. It is defined as the amount of heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius (°C).
This property varies for different substances. Water, for example, has a relatively high specific heat capacity of 4.18 J/g°C.

• This means water requires more energy to change its temperature compared to other substances with lower specific heat capacities.
• The specific heat capacity can affect how a substance is used in applications involving heat transfer, such as in cooling systems.

In our exercise, we use the water's specific heat capacity to determine how much the temperature of the water will increase when it absorbs heat released from burning the hydrogen and methane mixture.

Combustion Reactions

Combustion reactions are exothermic chemical reactions that occur when a substance reacts with an oxidant, releasing energy in the form of heat and light.
Typically involving a hydrocarbon and oxygen, combustion reactions fuel many everyday processes, including heating and powering engines.
When hydrogen and methane undergo combustion, they react with oxygen to produce water, carbon dioxide, and significant amounts of heat:

• These reactions are characterized by their high energy yields due to the release of enthalpy entailed in breaking molecular bonds.
• Both hydrogen and methane are valued as energy sources because they produce a lot of heat per gram during combustion.

The exercise provides a practical context for applying this concept by calculating the energy released from burning these fuels and transferring that heat to water.

Heat Transfer

Heat transfer is the process through which thermal energy is exchanged between different materials or systems.
It can occur by conduction, convection, or radiation, with conduction being the mode involved when heating water in this scenario.
Thermal energy will transfer from hot to cold regions until thermal equilibrium is reached:

• The specific heat capacity of water and the heat released from combustion reactions help us predict how much the water's temperature will rise.
• Heat transfer calculations are essential in ensuring energy efficiency, safety, and effectiveness in engineering and environmental applications.

During the exercise, we calculated how the total heat produced from hydrogen and methane combustion impacts the water's temperature, showing practical implications of heat transfer in everyday life.