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frmn
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- Homework Statement
- Consider a long cylindrical vessel of base area A_0. The ends of the vessel are fitted with two massless metallic plates also of area A_0. The bottom plate is fixed while the top plate is free to move. The initial length of the cylindrical column between the plates is L_0, in which n moles of an ideal gas (f=4) are filled, initially at a temperature T_0. It has a hypothetical permittivity ε(T)=k/T, and acts as an isotropic dielectric. The walls of the vessel are electrically insulating but allow heat exchange. The two metallic ends are connected to the terminals of an ideal battery of constant emf V_0. Initially, the upper plate is in equilibrium. The gas is then subjected to a polytropic process such that the process is quasi static, so as to expand until the length of the column is 2L_0. Thus, find the polytropic index a of the process, and also the heat exchange (if any).
- Relevant Equations
- PV=nRT
dU=TdS-dW
P_elec=σ²/2ε
C=Aε/L
What I'm able to do so far :
The magnitude of the charge on the plates as a function of length L and ε should be
Q=CV_0=A_0ε/L × V_0 = A_0 V_0 ε / L
So the force on the top plate should be
F_elec = QE = A_0 V_0² ε / 2 L²
So for the process to proceed quasi statically, the force due to pressure should be equal to F_elec
P A_0 = A_0 V_0² ε / 2 L²
P = V_0² ε / 2 L²
Now since V=A_0 L
P = A_0² V_0² k / 2 V² T
So we're getting
P V² T = constant
Since T = P V / n R ,
P² V³ = constant
So a = 1.5
So since L_final = 2 L_0
V_f = 2 V_0
As per the process T_final = T_0 / √2
Now ½ C V_0²
with C= A_0 ε / L should be the Helmholtz Free Energy
So the internal electromagnetic energy should be
U=F-T(dF/dT)
=½ A_0 V_0² / L [ ε - T dε/dT ]
= A_0 V_0² k / L T
So the total internal energy should be
U = ½ n f R T + A_0 V_0² k / L T
= 2 n R T + A_0 V_0² k / L T
So the net internal energy change should be
T : T_0 → T_0 / √2 , L : L_0 → 2 L_0
∆U = 2 n R T_0 [ 1/√2 -1 ] + A_0 V_0² k / L_0 T_0 [ 1/√2 -1 ]
Now if we use the initial equilibrium relation,
P_0 = V_0² k / 2 L_0² T_0
V_0 = A_0 L_0
P_0 V_0 = A_0 V_0 ² k / 2 L_0 T_0
P_0 V_0 = n R T_0
So we finally get
∆U = 4 n R T_0 [ 1/√2 -1 ]
Now to find ∆Q, taking the capacitor plates + the gas as our system,
∆U=∆Q+W_battery
W_battery = V_0 ∆q = V_0² ∆C
Upon calculating we again get
W_battery = 2 n R T_0 [ 1/√2 - 1 ]
So ∆Q= 2 n R T_0 [ 1/√2 -1 ]
Is this correct ? As I am not confident about the calculation of the internal energy of the dielectric.
In some places the mentioned expression for energy density is given as
u=½ E² [ ε + T dε/dT ]
But in this case I'm getting a negative sign.
Thanks in advance.
The magnitude of the charge on the plates as a function of length L and ε should be
Q=CV_0=A_0ε/L × V_0 = A_0 V_0 ε / L
So the force on the top plate should be
F_elec = QE = A_0 V_0² ε / 2 L²
So for the process to proceed quasi statically, the force due to pressure should be equal to F_elec
P A_0 = A_0 V_0² ε / 2 L²
P = V_0² ε / 2 L²
Now since V=A_0 L
P = A_0² V_0² k / 2 V² T
So we're getting
P V² T = constant
Since T = P V / n R ,
P² V³ = constant
So a = 1.5
So since L_final = 2 L_0
V_f = 2 V_0
As per the process T_final = T_0 / √2
Now ½ C V_0²
with C= A_0 ε / L should be the Helmholtz Free Energy
So the internal electromagnetic energy should be
U=F-T(dF/dT)
=½ A_0 V_0² / L [ ε - T dε/dT ]
= A_0 V_0² k / L T
So the total internal energy should be
U = ½ n f R T + A_0 V_0² k / L T
= 2 n R T + A_0 V_0² k / L T
So the net internal energy change should be
T : T_0 → T_0 / √2 , L : L_0 → 2 L_0
∆U = 2 n R T_0 [ 1/√2 -1 ] + A_0 V_0² k / L_0 T_0 [ 1/√2 -1 ]
Now if we use the initial equilibrium relation,
P_0 = V_0² k / 2 L_0² T_0
V_0 = A_0 L_0
P_0 V_0 = A_0 V_0 ² k / 2 L_0 T_0
P_0 V_0 = n R T_0
So we finally get
∆U = 4 n R T_0 [ 1/√2 -1 ]
Now to find ∆Q, taking the capacitor plates + the gas as our system,
∆U=∆Q+W_battery
W_battery = V_0 ∆q = V_0² ∆C
Upon calculating we again get
W_battery = 2 n R T_0 [ 1/√2 - 1 ]
So ∆Q= 2 n R T_0 [ 1/√2 -1 ]
Is this correct ? As I am not confident about the calculation of the internal energy of the dielectric.
In some places the mentioned expression for energy density is given as
u=½ E² [ ε + T dε/dT ]
But in this case I'm getting a negative sign.
Thanks in advance.