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Chapter 6: Problem 56

Hydrogen gives off \(120 . \mathrm{J} / \mathrm{g}\) of energy when burned in oxygen, and methane gives off \(50 .\) J/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

Expert verified
The final temperature of water can be found by following these steps: 1. Calculate the energy released by burning hydrogen: \(E_{H} = 5.0 \, g * 120 \, J/g\) 2. Calculate the energy released by burning methane: \(E_{M} = 10 \, g * 50 \, J/g\) 3. Calculate the total energy released: \(E_{T} = E_{H} + E_{M}\) 4. Calculate the temperature change using the specific heat capacity formula: 螖罢 = \(E_{T}/(50.0 \, g * 4.186 \, J/g^{\circ}C)\) 5. Calculate the final temperature: Final temperature (T2) = Initial temperature (T1) + Temperature change (螖罢)

Step by step solution

01

Calculate energy released by burning hydrogen.

To calculate the energy released, we will multiply the mass of hydrogen by the energy given off per gram, which is \(120 \, J/g\): Energy released by hydrogen = Mass of hydrogen 脳 Energy per gram \(E_{H} = 5.0 \, g * 120 \, J/g \)
02

Calculate energy released by burning methane.

Just like hydrogen, we will multiply the mass of methane by the energy given off per gram, which is \(50 \, J/g\): Energy released by methane = Mass of methane 脳 Energy per gram \(E_{M}= 10 \, g * 50 \, J/g\)
03

Calculate total energy released.

Add the energy released by the hydrogen and the energy released by the methane to find the total energy released: Total energy released = Energy released by hydrogen + Energy released by methane \(E_{T} = E_{H} + E_{M}\)
04

Use the specific heat capacity formula to find the temperature change.

The specific heat capacity formula is: Q = mc螖罢 Here, Q is the energy released or absorbed, m is the mass, c is the specific heat capacity, and 螖罢 is the temperature change. For water, the specific heat capacity c is \(4.186 \, J/g^{\circ}C\). We're given the mass of water m which is \(50.0 \, g\), and the initial temperature (T1) which is \(25.0^{\circ}C\). We can rewrite the formula for 螖罢: 螖罢 = Q / (mc) Substitute the values for total energy, mass, and specific heat capacity: 螖罢 = \(E_{T}/(50.0 \, g * 4.186 \, J/g^{\circ}C)\)
05

Calculate final temperature.

To find the final temperature (T2), we'll add the initial temperature to the temperature change: Final temperature (T2) = Initial temperature (T1) + Temperature change (螖罢) Now, we have all the required information to calculate the final temperature of water after transferring the heat released from burning hydrogen and methane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Released in Reactions
In thermochemistry, understanding the energy released in reactions is crucial. Chemical reactions involve the breaking and forming of bonds, and these processes either absorb or release energy, usually in the form of heat.

When substances like hydrogen and methane combust, they release energy. The amount of energy released per unit mass is expressed in joules per gram (J/g). The exercise deals with a mixture of hydrogen and methane, each releasing energy at different rates when burnt in oxygen. To calculate the total energy released, we multiply the mass of each substance by its respective energy release rate:
  • For hydrogen: Energy released = Mass of hydrogen x Energy per gram
  • For methane: Energy released = Mass of methane x Energy per gram

This sum gives us the total energy made available for heat transfer, significantly impacting the subsequent temperature rise in an absorbing substance, like water in this case.
Specific Heat Capacity
Specific heat capacity is a measure of how much energy (in joules) is needed to raise the temperature of one gram of a substance by one degree Celsius (掳C). It's a fundamental property intrinsic to each material.

Water has a high specific heat capacity at approximately 4.186 J/g掳C, meaning it can absorb a lot of heat with only a small rise in temperature. This unique characteristic allows water to act as a thermal buffer or reservoir, which is pivotal in the exercise we are examining. To influence the temperature of the water, the heat from the burned hydrogen and methane is absorbed, leading to a temperature change that is calculated using the formula for specific heat capacity:
Q = mc螖罢
where Q is the quantity of heat absorbed, m is the mass of the substance (water in this case), c is the specific heat capacity, and 螖罢 is the change in temperature.
Heat Transfer Calculations
The process of heat transfer calculations involves determining how much energy is transferred to or from a substance, causing a change in temperature. In our scenario, the energy released from burning fuels is transferred to water, which absorbs the heat.

Following the steps outlined in the textbook solution, step 4 utilizes the specific heat capacity concept to derive the change in temperature using the formula rearranged as 螖罢 = Q / (mc). This equation bridges the relationship between the released energy (Q) and the resultant temperature change (螖罢). By dividing the total energy by the product of the water's mass and its specific heat capacity, we find the temperature change.

Finally, knowing the initial water temperature and the calculated temperature change, we add these to find the water's final temperature. Through this process, students can clearly visualize how energy transfer functions in a real-world context, making the connection between abstract formulas and tangible phenomena.

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Most popular questions from this chapter

One of the components of polluted air is NO. It is formed in the high- temperature environment of internal combustion engines by the following reaction: $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}(g) \quad \Delta H=180 \mathrm{~kJ} $$ Why are high temperatures needed to convert \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) to NO?

Consider a mixture of air and gasoline vapor in a cylinder with a piston. The original volume is \(40 . \mathrm{cm}^{3} .\) If the combustion of this mixture releases 950. J of energy, to what volume will the gases expand against a constant pressure of 650 . torr if all the energy of combustion is converted into work to push back the piston?

Consider the reaction $$ \begin{array}{r} 2 \mathrm{HCl}(a q)+\mathrm{Ba}(\mathrm{OH})_{2}(a q) \longrightarrow \mathrm{BaCl}_{2}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l) \\ \Delta H=-118 \mathrm{~kJ} \end{array} $$ Calculate the heat when \(100.0 \mathrm{~mL}\) of \(0.500 \mathrm{M} \mathrm{HCl}\) is mixed with \(300.0 \mathrm{~mL}\) of \(0.100 \mathrm{M} \mathrm{Ba}(\mathrm{OH})_{2}\). Assuming that the temperature of both solutions was initially \(25.0^{\circ} \mathrm{C}\) and that the final mixture has a mass of \(400.0 \mathrm{~g}\) and a specific heat capacity of \(4.18 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}\), calculate the final temperature of the mixture.

A sample of nickel is heated to \(99.8^{\circ} \mathrm{C}\) and placed in a coffeecup calorimeter containing \(150.0 \mathrm{~g}\) water at \(23.5^{\circ} \mathrm{C}\). After the metal cools, the final temperature of metal and water mixture is \(25.0^{\circ} \mathrm{C}\). If the specific heat capacity of nickel is \(0.444 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}\), what mass of nickel was originally heated? Assume no heat loss to the surroundings.

The preparation of \(\mathrm{NO}_{2}(g)\) from \(\mathrm{N}_{2}(g)\) and \(\mathrm{O}_{2}(g)\) is an endothermic reaction: $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{NO}_{2}(g) \text { (unbalanced) } $$ The enthalpy change of reaction for the balanced equation (with lowest whole- number coefficients) is \(\Delta H=67.7 \mathrm{~kJ}\). If \(2.50 \times\) \(10^{2} \mathrm{~mL} \mathrm{~N}_{2}(g)\) at \(100 .{ }^{\circ} \mathrm{C}\) and \(3.50\) atm and \(4.50 \times 10^{2} \mathrm{~mL} \mathrm{O}_{2}(g)\) at \(100 .{ }^{\circ} \mathrm{C}\) and \(3.50 \mathrm{~atm}\) are mixed, what amount of heat is necessary to synthesize the maximum yield of \(\mathrm{NO}_{2}(g)\) ?
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