A monoatomic gas at a pressure P having a volume v expands isothermally to a volume 3V and then compresses adiabatically to a volume v the final pressure of the gas is.
(only the answers which satisfy me will be given pts)
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A monoatomic gas at a pressure P having a volume v expands isothermally to a volume 3V and then compresses adiabatically to a volume v the final pressure of the gas is.
(only the answers which satisfy me will be given pts)
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Answer:
To solve this problem, we can use the ideal gas law and the equations for isothermal and adiabatic processes.
Let's denote the initial pressure, volume, and final pressure as P₁, V₁, and P₂, respectively.
Isothermal Expansion:
In an isothermal process, the temperature remains constant. The equation for an isothermal process is:
P₁V₁ = P₂V₂
Given that the volume expands from V₁ to 3V (V₂ = 3V), we can rewrite the equation as:
P₁V₁ = P₂(3V)
Adiabatic Compression:
In an adiabatic process, there is no heat exchange with the surroundings, which implies that the equation PV^γ = constant applies, where γ is the heat capacity ratio (specific heat capacity at constant pressure divided by specific heat capacity at constant volume).
For a monoatomic ideal gas, γ = 5/3.
Using this equation, we can express the adiabatic compression as:
P₂V₂^γ = P₃V₃^γ
Since the final volume is V₁ (compresses back to the original volume), we have V₃ = V₁.
Combining the equations for the isothermal and adiabatic processes:
P₁V₁ = P₂(3V)
P₂V₁^(5/3) = P₃V₁^(5/3)
Dividing these two equations, we can eliminate V₁:
(P₁V₁) / (P₂V₁^(5/3)) = (P₂(3V)) / (P₃V₁^(5/3))
Simplifying:
P₁ / P₂V₁^(2/3) = 3P₂ / (P₃V₁^(2/3))
P₁ / P₂ = 3P₂ / P₃
Cross-multiplying:
P₁P₃ = 3P₂^2
Finally, we can express the final pressure P₃ in terms of the given variables:
P₃ = (3P₂^2) / P₁
Therefore, the final pressure of the gas after adiabatic compression is
[tex] \sf{( \frac{(3P₂ {}^{2} )}{P₁} .}[/tex]
Answer: et's denote the initial pressure as
�
P, the initial volume as
�
V, and the final volume as
3
�
3V.
Isothermal Expansion:
During an isothermal process, the temperature remains constant. For an ideal gas, the relationship between pressure (
�
P), volume (
�
V), and temperature (
�
T) is given by the combined gas law:
�
�
=
�
�
�
PV=nRT
Since the process is isothermal, the temperature remains constant, and the equation becomes:
�
1
�
1
=
�
2
�
2
P
1
V
1
=P
2
V
2
where
�
1
P
1
and
�
1
V
1
are the initial pressure and volume, and
�
2
P
2
and
�
2
V
2
are the final pressure and volume.
�
⋅
�
=
�
′
⋅
3
�
P⋅V=P
′
⋅3V
Adiabatic Compression:
During an adiabatic process, there is no heat exchange with the surroundings, and the relationship between pressure (
�
P), volume (
�
V), and adiabatic index (
�
γ) is given by:
�
�
�
=
constant
PV
γ
=constant
For a monoatomic ideal gas,
�
=
5
3
γ=
3
5
.
During the adiabatic compression, we can use the fact that
�
2
�
2
�
=
�
3
�
3
�
P
2
V
2
γ
=P
3
V
3
γ
, where
�
3
P
3
and
�
3
V
3
are the final pressure and volume after adiabatic compression.
�
′
⋅
3
�
5
/
3
=
�
3
⋅
�
5
/
3
P
′
⋅3V
5/3
=P
3
⋅V
5/3
Now, let's solve these equations to find the final pressure
�
3
P
3
:
From the isothermal expansion:
�
⋅
�
=
�
′
⋅
3
�
P⋅V=P
′
⋅3V
�
′
=
�
3
P
′
=
3
P
Substitute
�
′
P
′
into the adiabatic compression equation:
�
3
⋅
3
�
5
/
3
=
�
3
⋅
�
5
/
3
3
P
⋅3V
5/3
=P
3
⋅V
5/3
�
3
=
�
3
P
3
=
3
P
So, the final pressure of the gas after the adiabatic compression is
�
3
3
P
.
Explanation: