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 final temperature, assuming the gas to be an ideal gas for which γ = 1.40 (c) How much work was done on the gas?

To solve this problem, we need to apply the principles of thermodynamics, particularly focusing on the adiabatic process for an ideal gas. In an adiabatic process, there is no heat exchange with the surroundings, and we can use specific equations that relate pressure, volume, and temperature. Let's break this down step by step.

Given Information

• Initial Volume (V1) = 4.33 L
• Initial Pressure (P1) = 1.17 atm
• Initial Temperature (T1) = 310 K
• Final Volume (V2) = 1.06 L
• Adiabatic index (γ) = 1.40

Finding Final Pressure (P2)

For an adiabatic process, we can use the following relationship between pressure and volume:

P1 * V1^γ = P2 * V2^γ

Rearranging this equation to solve for P2 gives us:

P2 = P1 * (V1/V2)^γ

Now, substituting the known values:

P2 = 1.17 atm * (4.33 L / 1.06 L)^1.40

Calculating the volume ratio:

(4.33 / 1.06) = 4.08

Now raising this to the power of 1.40:

(4.08)^1.40 ≈ 7.77

Now substituting back into the equation for P2:

P2 ≈ 1.17 atm * 7.77 ≈ 9.09 atm

Calculating Final Temperature (T2)

Next, we can find the final temperature using the relationship:

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)^(1.40 - 1)

Calculating the exponent:

(1.40 - 1) = 0.40

Now calculating the volume ratio raised to 0.40:

(4.08)^0.40 ≈ 2.36

Now substituting back into the equation for T2:

T2 ≈ 310 K * 2.36 ≈ 731.6 K

Work Done on the Gas

To find the work done on the gas during an adiabatic process, we can use the formula:

W = (P2 * V2 - P1 * V1) / (γ - 1)

Substituting the values we have:

W = (9.09 atm * 1.06 L - 1.17 atm * 4.33 L) / (1.40 - 1)

Calculating the pressures and volumes:

W = (9.63 - 5.06) / 0.40

W = 4.57 / 0.40 ≈ 11.43 atm·L

To convert atm·L to Joules (since 1 atm·L = 101.325 J), we multiply:

W ≈ 11.43 atm·L * 101.325 J/atm·L ≈ 1155.5 J

Summary of Results

• Final Pressure (P2): 9.09 atm
• Final Temperature (T2): 731.6 K
• Work Done on the Gas: 1155.5 J

This analysis illustrates how the properties of an ideal gas change during an adiabatic process, allowing us to calculate the final state of the gas and the work done on it. If you have any further questions or need clarification on any of the steps, feel free to ask!