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:
- 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.
- 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.