A monatomic ideal gas that is initially at 1.50×1051.50\$$times10^5 Pa $$and has a volume of 0.08000.0800 m3 is compressed adiabatically to a volume of 0.04000.0400 m3. What is the final pressure?

Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 19: The First Law of Thermodynamics

문제 27a

19장, 문제 27a

A monatomic ideal gas that is initially at $1.50 \times 10^{5}$ Pa and has a volume of $0.0800$ m³ is compressed adiabatically to a volume of $0.0400$ m³. What is the final pressure?

검증된 단계별 안내

Understand that the process is adiabatic, meaning no heat is exchanged with the surroundings. For an adiabatic process involving an ideal gas, the relation between pressure and volume is given by the equation: $ P_1 V_1^\gamma = P_2 V_2^\gamma $, where $ \gamma $ is the adiabatic index (ratio of specific heats, $ C_p/C_v $). For a monatomic ideal gas, $ \gamma = \frac{5}{3} $.

Identify the known values: initial pressure $ P_1 = 1.50 \times 10^5 $ Pa, initial volume $ V_1 = 0.0800 $ m$^3$, and final volume $ V_2 = 0.0400 $ m$^3$. The final pressure $ P_2 $ is what we need to find.

Substitute the known values into the adiabatic equation: $ 1.50 \times 10^5 \times (0.0800)^{\frac{5}{3}} = P_2 \times (0.0400)^{\frac{5}{3}} $.

Solve for $ P_2 $ by isolating it on one side of the equation: $ P_2 = \frac{1.50 \times 10^5 \times (0.0800)^{\frac{5}{3}}}{(0.0400)^{\frac{5}{3}}} $.

Calculate the expression to find the final pressure $ P_2 $. Ensure to handle the powers and division correctly to arrive at the numerical value for $ P_2 $.

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주요 개념

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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, this means that any change in internal energy is due solely to work done on or by the system. In an adiabatic compression, the gas's temperature and pressure increase as the volume decreases.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, and temperature of an ideal gas through the equation 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 in understanding how gases behave under different conditions, although it assumes no interactions between gas molecules.

Adiabatic Equation for Ideal Gases

For an adiabatic process involving an ideal gas, the relationship between pressure and volume is given by the equation PV^γ = constant, where γ (gamma) is the heat capacity ratio (Cp/Cv). This equation allows us to calculate the final pressure of the gas after adiabatic compression or expansion, given the initial conditions and the final volume.