Question

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 \(950 .\) \(\mathrm{J}\) 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?

Short answer

The volume to which the gases expand against a constant pressure of 650 torr after the combustion of the mixture, which releases 950 J of energy, is approximately 11000 cm³.

Step-by-step solution

Convert pressure to the SI unit

Since the given pressure is in torr, we first need to convert it to the SI unit of pressure, which is pascal (Pa). We use the following conversion factor: 1 torr = 133.32 Pa. \(P_{torr} = 650 \, \text{torr}\) \( P = P_{torr} \times 133.32 \, \frac{\text{Pa}}{\text{torr}} \) \( P = 650 \, \text{torr} \times 133.32 \, \frac{\text{Pa}}{\text{torr}} \) \( P \approx 86658 \, \text{Pa} \) So the pressure in pascal is approximately 86658 Pa.

Calculate work done by the system

Now we need to find the work done by the system against the constant external pressure. We are given that the energy released due to combustion is completely converted into work. Therefore, work = 950 J.

Calculate the change in volume

We can find the change in volume (Δvolume) using the work done formula: Work = Pressure × Δvolume Δvolume = Work ÷ Pressure Δvolume = 950 J ÷ 86658 Pa Δvolume ≈ 0.01096 m³ Note that we have calculated the change in volume in cubic meters (m³). We will convert it back to cubic centimeters (cm³) in the final step.

Calculate the final volume

Now, we have the initial volume and the change in volume. We can find the final volume by adding the change in volume to the initial volume: Initial volume = 40 cm³ = 0.00004 m³ (after converting to m³) Final volume = Initial volume + Δvolume Final volume = 0.00004 m³ + 0.01096 m³ Final volume ≈ 0.01100 m³

Convert the final volume back to cubic centimeters

To convert the final volume back to cubic centimeters: Final volume = 0.01100 m³ × (100 cm/m)³ Final volume ≈ 11000 cm³ So, the volume to which gases expand is approximately 11000 cm³.

Key concepts

Chemical Energy

Chemical energy is a form of potential energy stored within chemical bonds. It's a vital concept in chemistry, especially when discussing combustion reactions, where this stored energy is released as heat and light.

In our exercise, the chemical energy of the gasoline vapor is the source of the 950 joules of heat energy released during the combustion process. Understanding chemical energy is essential because it dictates how much work can be performed by the system. This energy is initially locked within the molecular structure of the gasoline and, when the combustion reaction occurs, it is released and converted into work to push the piston.

The conversion of chemical energy into work is at the heart of many technological applications, such as internal combustion engines that power cars. It is also linked to the law of conservation of energy, which states that energy cannot be created or destroyed but can be transformed from one form to another.

Gas Laws

The gas laws are a series of laws that describe the behavior of gases under varying conditions of temperature, volume, and pressure. They are critical when predicting how a gas will respond to certain conditions, like those seen in the cylinder of an engine during combustion.

Boyle's Law and Work

One of the gas laws relevant to our exercise is Boyle's Law, which indicates that the volume of a gas varies inversely with its pressure, provided the temperature and mass remain constant. When the combustion reaction releases energy and work is done on the piston, Boyle's Law helps explain the expansion of the gases.

In the exercise, we're dealing with a constant pressure; thus, the direct relationship between work (energy transferred when the gas expands) and volume change showcases principles closely related to Boyle's Law. Understanding how pressure and volume interact is crucial for solving real-world problems involving gases, including the design of combustion engines and predicting the behavior of the atmosphere.

Thermochemistry

Thermochemistry is the study of the heat energy associated with chemical reactions and physical transformations. It is a branch of thermodynamics that deals with the relationship between chemical reactions and energy changes involving heat.

Energy Conversion

Thermochemistry principles enable us to understand how the energy from the gasoline's combustion can perform work on the piston in the cylinder. The exercise showcases the first law of thermodynamics or the law of energy conservation, where the energy released during combustion is fully converted into work.

By observing that the chemical energy released (950 J) is equal to the work done by the gas upon expansion, we apply the concept of enthalpy—a measure of the total heat content in a system. In an isolated system, all the heat produced from the chemical reaction (combustion) would be equal to the work done because there's no heat exchange with the surroundings, which aligns with our exercise scenario.

Thermochemistry is essential in many fields, from environmental science, where it helps evaluate the heat exchange in our ecosystem, to engineering, where it influences the design of energy-efficient systems and engines.