A heat engine uses a diatomic gas that follows the pV cycle in FIGURE P21.59. Determine the pressure, volume, and temperature at point 2.

Name: Physics for Scientists and Engineers
Author: Knight Calc
ISBN: 9780137344796

검증된 단계별 안내

Step 1: Identify the given information from the graph. At point 1, the pressure (p₁) is 400 kPa, the volume (V₁) is 2000 cm³, and the temperature (T₁) is 300 K. The process from point 1 to point 2 is adiabatic, meaning no heat is exchanged.

Step 2: Use the adiabatic relation for a diatomic gas, which is governed by the equation p₁V₁^γ = p₂V₂^γ, where γ (gamma) is the adiabatic index. For a diatomic gas, γ = 7/5. Rearrange this equation to solve for p₂.

Step 3: Determine the volume at point 2 (V₂) from the graph. From the figure, V₂ is approximately 1000 cm³. Substitute V₂, p₁, V₁, and γ into the adiabatic equation to calculate p₂.

Step 4: Use the ideal gas law, pV = nRT, to find the temperature at point 2 (T₂). Rearrange the equation to T₂ = (p₂V₂)/(nR). Substitute the values of p₂, V₂, and the gas constant R (8.314 J/mol·K) to calculate T₂. Note that n (number of moles) can be determined if additional information is provided.

Step 5: Summarize the results. You now have the pressure (p₂), volume (V₂), and temperature (T₂) at point 2. Ensure all units are consistent (e.g., convert cm³ to m³ for volume if necessary).

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Ideal Gas Law

The Ideal Gas Law relates the pressure (P), volume (V), and temperature (T) of an ideal gas through the equation PV = nRT, where n is the number of moles and R is the ideal gas constant. This law is fundamental for understanding the behavior of gases under varying conditions and is essential for calculating the state variables at different points in a thermodynamic cycle.

Thermodynamic Cycles

A thermodynamic cycle is a series of processes that return a system to its initial state, allowing for the analysis of energy transfer and work done. In the context of heat engines, these cycles often involve isothermal, adiabatic, isochoric, and isobaric processes, which help in understanding how energy is converted from heat to work and vice versa.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings, meaning all changes in internal energy are due to work done on or by the system. For a diatomic gas, the relationship between pressure, volume, and temperature during an adiabatic process can be described by specific equations, which are crucial for determining the state of the gas at various points in the cycle.