Wize University Physics Textbook (Master) > Thermodynamics

PV- Diagrams

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PV Diagrams & Thermodynamic Processes

  • The system is taken from some initial state to some final state.
  • Every line in a PV-diagram is a thermodynamic process.
  • There are four well known thermodynamic processes


Image illustrating PV Diagrams & Thermodynamic Processes
  • The adiabatic line is always steeper than the isothermal line because:
  • In an isothermal process: P=nRTV∝1VP=\frac{nRT}{V}\propto\frac{1}{V}
  • In an adiabatic process: P=ConstantVγ∝1Vγ, γ>1P=\frac{Constant}{V^\gamma} \propto\frac{1}{V^\gamma},\ \gamma>1


Work done on an ideal gas

Work done on the gas during a thermodynamic process is the area under the PV curve, from initial state (i) to final state (f).

Won the gas=∫PdVW_{on \ the\ gas} =\int PdV


EXPANSION of the gas COMPRESSION of the gas
Image illustrating PV Diagrams & Thermodynamic Processes


Image illustrating PV Diagrams & Thermodynamic Processes



A Carnot engine is operated between ��H and ��C according to the PV – Diagram given below where the processes 𝑎 → 𝑏 and 𝑐 → 𝑑 are isothermal and the processes 𝑏 → 𝑐 and 𝑑 → 𝑎 are adiabatic.



Image illustrating Example: PV Diagram


a) Find the efficiency of this engine as a function of ��C and ��H

b) Show Carnot Cycle in TS –diagram


Solution:


Image illustrating Example: PV Diagram

a) Qbc=Qda=0 adiabaticQ_{bc}=Q_{da}=0\ adiabatic


Qab:ΔT=0→Wab=Qab>0Q_{ab}:\Delta T=0\to W_{ab}=Q_{ab}>0


Qab=nRTHln⁡(VbVa)Q_{ab}=nRT_H\ln\left(\frac{V_b}{V_a}\right)


Qcd=Wcd<0→Qcd=nRTc ln⁡(VdVc)Q_{cd}=W_{cd}<0\to Q_{cd}=nRT_c\ \ln\left(\frac{V_d}{V_c}\right)


e=1+QcQH=1+TcTH ln⁡(VdVc)ln⁡(VbVa)e=1+\frac{Q_c}{Q_H}=1+\frac{T_c}{T_H}\ \frac{\ln\left(\frac{V_d}{V_c}\right)}{\ln\left(\frac{V_b}{V_a}\right)}


TVγ−1=constTV^{\gamma-1}=const for adiabatic processes


for b→c: THVbγ−1=TcVcγ−1→THTC=(VcVb)γ−1b\to c:\ T_HV_b^{\gamma-1}=T_cV_c^{\gamma-1}\to\frac{T_H}{T_C}=\left(\frac{V_c}{V_b}\right)^{\gamma-1}


for d→a: TCVdγ−1=THVaγ−1→THTC=(VdVa)γ−1d\to a:\ T_CV_d^{\gamma-1}=T_HV_a^{\gamma-1}\to\frac{T_H}{T_C}=\left(\frac{V_d}{V_a}\right)^{\gamma-1}


→VcVb=VdVa or VdVc=VaVb\to\frac{V_c}{V_b}=\frac{V_d}{V_a}\ or\ \frac{V_d}{V_c}=\frac{V_a}{V_b} plug into efficiency equation


→e=1−TcTH\to e=1-\frac{T_c}{T_H}

b)

Image illustrating Example: PV Diagram


0.08 mole of nitrogen undergoes the following process. What is the work done on the gas?

Illustration for practice question: First Law of Thermodynamics



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Mark Yourself Question

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Two moles of a monatomic gas undergo some thermodynamic processes as shown in the following plot. (The expansion from a to b is adiabatic). What is the net work done by the gas during a full cycle? (As a function of ��𝑜 and 𝑉𝑜)

Illustration for practice question: Thermodynamic Processes



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Extra Practice

New Practice Question

An engineer wants to design! A thermodynamic path in order to increase the temperature of 1000 lit of a diatomic gas from 350K to 850K. He is considering two different options. The first method is to increase the temperature using a series of alternating adiabatic and isothermal processes so that during each adiabatic process the gas is heated up by 100K and then is expanded to its original volume through an isothermal process. The second method is to raise the temperature through a single isochoric process.

a) Draw P-V and T-S curves for both methods

b) In which method the heat added to the gas is less?

PV and TS Diagrams

A system of a cylinder and a piston has the shape of the diagram below when the temperature of the gas inside the cylinder is 300℃ and its pressure is 200𝐾𝑃𝑎. The piston has a mass of 50 𝑘𝑔 and its radius is . 06 𝑚 and pushes against a set of stops at initial condition so that the height of the cylinder is 80 𝑐𝑚. The temperature outside the cylinder is 30℃ and the pressure is 1 𝑎𝑡𝑚. The cylinder cools down by exchanging heat with the environment.

a. Draw a free body diagram of the piston.

b. At what temperature and pressure does the piston start to move down?

A system of a cylinder and a piston has the shape of the diagram below when the temperature of the gas inside the cylinder is 300℃ and its pressure is 200𝐾𝑃𝑎. The piston has a mass of 50 𝑘𝑔 and its radius is . 06 𝑚 and pushes against a set of stops at initial condition so that the height of the cylinder is 80 𝑐𝑚. The temperature outside the cylinder is 30℃ and the pressure is 1 𝑎𝑡𝑚. The cylinder cools down by exchanging heat with the environment.

a. Draw a free body diagram of the piston.

b. At what temperature and pressure does the piston start to move