Gas occupies ta volume of 4.33 L at a pressure of 1.17 atm and temperature of 310 K it is compressed adiabatically to a volume of 1.06 L,. Determine (a) the final pressure and (b) the

To solve the problem of gas compression under adiabatic conditions, we can use the principles of thermodynamics, specifically the adiabatic process equations. In an adiabatic process, there is no heat exchange with the surroundings, and we can apply the ideal gas law and the adiabatic relations to find the final pressure and temperature after compression.

Understanding Adiabatic Processes

In an adiabatic process, the relationship between pressure and volume can be expressed using the formula:

• P1 * V1^γ = P2 * V2^γ

Here, P1 and V1 are the initial pressure and volume, P2 and V2 are the final pressure and volume, and γ (gamma) is the heat capacity ratio (C_p/C_v) for the gas. For many diatomic gases like nitrogen and oxygen, γ is approximately 1.4, while for monatomic gases like helium, it is about 1.67. For this example, we will assume the gas behaves like a diatomic gas, so γ = 1.4.

Step-by-Step Calculation

Given Values

• Initial Volume (V1) = 4.33 L
• Initial Pressure (P1) = 1.17 atm
• Final Volume (V2) = 1.06 L

Finding Final Pressure (P2)

Using the adiabatic relation, we can rearrange the formula to solve for P2:

• P2 = P1 * (V1/V2)^γ

Now, substituting the known values:

• P2 = 1.17 atm * (4.33 L / 1.06 L)^1.4

Calculating the volume ratio:

• 4.33 L / 1.06 L ≈ 4.09

Now raising this ratio to the power of γ:

• (4.09)^1.4 ≈ 7.56

Now substituting back into the equation for P2:

• P2 ≈ 1.17 atm * 7.56 ≈ 8.84 atm

Final Pressure Result

The final pressure after the adiabatic compression is approximately 8.84 atm.

Determining Final Temperature

To find the final temperature (T2), we can use the ideal gas law and the relationship between temperature and pressure during an adiabatic process:

• T1 * V1^(γ-1) = T2 * V2^(γ-1)

Rearranging gives us:

• T2 = T1 * (V1/V2)^(γ-1)

Substituting the known values:

• T2 = 310 K * (4.33 L / 1.06 L)^(1.4 - 1)

Calculating the exponent:

• 1.4 - 1 = 0.4

Now calculating the volume ratio raised to the power of 0.4:

• (4.09)^0.4 ≈ 2.03

Now substituting back into the equation for T2:

• T2 ≈ 310 K * 2.03 ≈ 630.93 K

Final Temperature Result

The final temperature after the adiabatic compression is approximately 630.93 K.

Summary of Results

In summary, after compressing the gas adiabatically from a volume of 4.33 L to 1.06 L:

• Final Pressure (P2) ≈ 8.84 atm
• Final Temperature (T2) ≈ 630.93 K

This process illustrates how gas behavior changes under different conditions, and understanding these principles is crucial in fields like thermodynamics and engineering.