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: