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. 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 gases will expand to a volume of approximately \(10940 cm^3\) after all the energy from combustion has been converted into work to push back the piston.

Step-by-step solution

Convert energy to work

We are given that the combustion of the mixture releases 950 J of energy, and we will assume that all the energy is converted into work to push back the piston. Work = Energy released by combustion So, Work = 950 J

Convert pressure to Pascals

We are given a constant pressure of 650 torr. We need to convert this to Pascals (Pa) to use it further in our calculations. 1 atm = 760 torr 1 atm = 101325 Pa Pressure (Pa) = (Pressure (torr) * 1 atm) / 760 torr * 101325 Pa/atm Pressure (Pa) = (650 * 1 atm) / 760 * 101325 Pa/atm ≈ 87277 Pa

Calculate work done against constant pressure

We can calculate the work done against the constant pressure using the following formula: Work = Pressure × Change in Volume We know the Work (950 J) and the Pressure (87277 Pa), so we can calculate the Change in Volume: Change in Volume = Work / Pressure Change in Volume ≈ 950 J / 87277 Pa ≈ 0.0109 m^3

Convert original volume to m^3

We are given the original volume in cm^3, and we need to convert it to m^3 to be consistent with the units used above. Original Volume (m^3) = Original Volume (cm^3) * (1 m / 100 cm)^3 Original Volume (m^3) = 40 cm^3 * (1 m / 100 cm)^3 ≈ 4.0 × 10^-5 m^3

Calculate the final volume

Now we can add the Change in Volume to the Original Volume to find the Final Volume: Final Volume (m^3) = Original Volume (m^3) + Change in Volume (m^3) Final Volume (m^3) ≈ 4.0 × 10^-5 m^3 + 0.0109 m^3 ≈ 0.01094 m^3

Convert final volume back to cm^3

Finally, we will convert the Final Volume back to cm^3 to match the given units: Final Volume (cm^3) = Final Volume (m^3) * (100 cm / 1 m)^3 Final Volume (cm^3) ≈ 0.01094 m^3 * (100 cm / 1 m)^3 ≈ 10940 cm^3 The gases will expand to a volume of approximately 10940 cm^3 after all the energy from combustion has been converted into work to push back the piston.

Key concepts

Combustion

Combustion is a chemical reaction that occurs when a substance reacts rapidly with oxygen and releases energy in the form of heat and light. It lies at the heart of many mechanical processes, especially in engines like car engines. For the mixture of air and gasoline vapor in our problem, combustion provides the necessary energy to do work.
This energy from combustion can be harnessed to move pistons in engines, enabling machines to perform tasks.

• Combustion involves fuels reacting with oxygen to produce exhaust gases and energy.
• In our context, the energy released during combustion is 950 J.
• This energy can be converted into work, causing mechanical movement.

The efficiency of such energy conversion is crucial in designing engines and systems that maximize the energy output from combustion, consequently improving performance and reducing fuel consumption.

Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in energy of that object. In simpler terms, energy is transferred to achieve work.
In our exercise, energy derived from combustion is used entirely to perform work. This energy transformation is crucial for understanding how engines function.

• Work is calculated using the formula: \( \text{Work} = \text{Pressure} \times \text{Change in Volume} \).
• The entire energy of 950 J released is assumed to be converted into work.
• Through this conversion, the moving piston creates increased space for gas expansion.

The work-energy principle is a fundamental concept in thermodynamics and mechanical engineering, illustrating how energy conversion facilitates mechanical processes.

Pressure Conversion

Pressure conversion is vital when dealing with different measurement units. It allows for accurate calculations, especially in physics, where consistent units are necessary. In our problem, this involves converting atmospheric pressure from torr to Pascals (Pa).

• Pressure given is 650 torr and needs conversion to Pascals to be used in calculations.
• The conversion factor used is based on 1 atm equaling 760 torr and 101325 Pa.
• Resultantly, 650 torr converts to approximately 87277 Pascals.

Selecting the appropriate pressure unit is essential for maintaining the integrity of calculations, ensuring accurate and meaningful results in thermodynamics and fluid dynamics.

Gas Expansion

Gas expansion describes the increase in volume that gases undergo when energy is applied to them, caused by increased molecular motion. This concept is evident in many thermodynamic applications, especially when a piston is used to control volume changes within a container.

• In our exercise, the combustion energy leads to an increase in gas volume.
• Initial volume was given in cm³, which needed conversion and calculations to find the final expanded volume.
• From an original volume of 40 cm³, gases expanded to almost 10940 cm³.

Understanding gas expansion is key for designing engines and compressors. It influences how systems respond to energy changes, which can inform engineering decisions to optimize performance and efficiency.