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Ch 19: The First Law of Thermodynamics
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 19: The First Law of ThermodynamicsProblema 28b
Capitolo 19, Problema 28b

The engine of a Ferrari F355 F1 sports car takes in air at 20.020.0°C and 1.001.00 atm and compresses it adiabatically to 0.09000.0900 times the original volume. The air may be treated as an ideal gas with g=1.40g = 1.40. Find the final temperature and pressure.

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Understand the concept of adiabatic processes: In an adiabatic process, no heat is exchanged with the surroundings. For an ideal gas undergoing an adiabatic process, the relationship between pressure, volume, and temperature is governed by the adiabatic condition: \( PV^\gamma = \text{constant} \), where \( \gamma \) is the adiabatic index.
Use the adiabatic relation for temperature: The relation between initial and final temperatures in an adiabatic process is given by \( T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{\gamma - 1} \). Here, \( T_1 \) is the initial temperature, \( V_1 \) is the initial volume, \( V_2 \) is the final volume, and \( \gamma = 1.40 \). Convert the initial temperature from Celsius to Kelvin by adding 273.15.
Calculate the final temperature: Substitute the given values into the temperature relation. \( T_1 = 20.0 + 273.15 \) K, \( V_2 = 0.0900 \times V_1 \), and \( \gamma = 1.40 \). Compute \( T_2 \) using the formula \( T_2 = T_1 \left( \frac{1}{0.0900} \right)^{0.40} \).
Use the adiabatic relation for pressure: The relation between initial and final pressures in an adiabatic process is given by \( P_2 = P_1 \left( \frac{V_1}{V_2} \right)^{\gamma} \). Here, \( P_1 = 1.00 \) atm is the initial pressure.
Calculate the final pressure: Substitute the given values into the pressure relation. Compute \( P_2 \) using the formula \( P_2 = 1.00 \left( \frac{1}{0.0900} \right)^{1.40} \). This will give you the final pressure in atm.

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Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is exchanged with the surroundings. For an ideal gas undergoing an adiabatic process, the relationship between pressure, volume, and temperature is governed by the adiabatic condition: PV^γ = constant, where γ (gamma) is the heat capacity ratio (Cp/Cv). This concept is crucial for determining the final temperature and pressure of the gas after compression.
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Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics, expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law helps relate the macroscopic properties of gases and is essential for calculating changes in state variables when the gas is treated as ideal, as in this problem.
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Heat Capacity Ratio (γ)

The heat capacity ratio, γ (gamma), is the ratio of the heat capacity at constant pressure (Cp) to the heat capacity at constant volume (Cv). It is a critical parameter in adiabatic processes, affecting how pressure and temperature change with volume. For diatomic gases like air, γ is typically around 1.40, which influences the calculations of final temperature and pressure in adiabatic compression.
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