Answer:
(ii) The final pressure (Pf) is related to the initial pressure (Pi) by the equation Pf = Pi * (Vi/Vf)^γ.
Step-by-step explanation:
When an ideal gas is kept inside a cylinder by a piston and then expands adiabatically, we can compare the initial (i) and final (f) physical quantities of the gas to each other.
In an adiabatic process, there is no heat exchange between the system and its surroundings. Therefore, the first law of thermodynamics can be simplified to:
ΔU = W,
where ΔU is the change in internal energy of the gas and W is the work done on or by the gas.
For an adiabatic expansion, the work done by the gas is negative, meaning the gas does work on its surroundings. This causes a decrease in the internal energy of the gas. Therefore, we can conclude that:
(i) The final internal energy of the gas (Uf) is less than the initial internal energy (Ui), i.e., Uf < Ui.
Since the work done by the gas is negative, we can relate it to the change in volume of the gas. The work done on the gas can be given by:
W = P(Vf - Vi),
where P is the pressure and V is the volume of the gas.
In an adiabatic process, the pressure and volume are related by the equation:
P(Vi)^γ = P(Vf)^γ,
where γ is the heat capacity ratio, which depends on the specific gas.
From this equation, we can rearrange to solve for the final pressure (Pf):
Pf = Pi * (Vi/Vf)^γ,
where Pi is the initial pressure.
So, in summary:
iii) The final pressure (Pf) is related to the initial pressure (Pi) by the equation Pf = Pi * (Vi/Vf)^γ.
By comparing the initial and final physical quantities, we can conclude that the final pressure (Pf) is related to the initial pressure (Pi) by the equation Pf = Pi * (Vi/Vf)^γ, and the final internal energy (Uf) is less than the initial internal energy (Ui), i.e., Uf < Ui.