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Which is not correct for an adiabatic process:
1. \(P_1^{1/\gamma }V_1=P_2^{1/\gamma }V_2\)
2. \(T_1V_1^{\gamma-1 }=T_2V_2^{\gamma-1 }\)
3. \(P_1^{1-\gamma}T_1^\gamma =P_2^{1-\gamma}T_2^\gamma \)
4. \(T_1V_1^{1-\gamma}=T_2V_2^{1-\gamma }\)

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Correct Answer - Option 4 : \(T_1V_1^{1-\gamma}=T_2V_2^{1-\gamma }\)

CONCEPT:

Adiabatic process:

  • When a thermodynamic system undergoes a change in such a way that no exchange of heat takes place between the System and surroundings, the process is known as the adiabatic process.


Essential conditions for the adiabatic process:

  1. There should not be any exchange of heat between the system and its surroundings. All walls of the container and the piston must be perfectly insulating.
  2. The system should be compressed or allowed to expand suddenly so that there is no time for the exchange of heat between the system and its surroundings.

 

  • The relation between the pressure and volume in an adiabatic process is given as,


⇒ PVγ = constant

Where P = pressure, V = volume, and γ = adiabatic index

EXPLANATION:

  • The ideal gas equation is given as,


⇒ PV = nRT     -----(1)

Where P = pressure, V = volume, T = temperature, n = number of moles, and R = gas constant

  • The relation between the pressure and volume in an adiabatic process is given as,


⇒ PVγ = constant     -----(2)

By equation 1 and equation 2,

\(\Rightarrow P_1^{1/\gamma }V_1=P_2^{1/\gamma }V_2\)     -----(3)

\(\Rightarrow T_1V_1^{\gamma-1 }=T_2V_2^{\gamma-1 }\)     -----(4)

\(\Rightarrow P_1^{1-\gamma}T_1^\gamma =P_2^{1-\gamma}T_2^\gamma \)    -----(5)

  • Hence, option 4 is correct.

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