Question

Carbon dioxide is compressed adiabatically in a piston-cylinder arrangement form 400 kPa and 310 K to 2 MPa. What is the minimum work required for the compression process if the initial volume is 0.04

Answer 1

To calculate the minimum work required for the adiabatic compression process of carbon dioxide, we can use the First Law of Thermodynamics, which states that:

ΔU = Q - W

Where:

ΔU = Change in internal energy

Q = Heat transfer

W = Work done on the gas

Since the process is adiabatic (Q = 0), there is no heat transfer. Therefore, the equation becomes:

ΔU = -W

The change in internal energy for an ideal diatomic gas during adiabatic compression is given by:

ΔU = (γ - 1) * n * R * (T2 - T1)

Where:

γ = Specific heat ratio for CO2 (approximately 1.4)

n = Number of moles of CO2

R = Universal gas constant (8.314 J/(mol K))

T2 = Final temperature (in Kelvin)

T1 = Initial temperature (in Kelvin)

Now, we need to find the final temperature (T2) using the adiabatic process equation:

P1 * V1^γ = P2 * V2^γ

Where:

P1 = Initial pressure (400 kPa = 400,000 Pa)

V1 = Initial volume (0.04 m^3)

P2 = Final pressure (2 MPa = 2,000,000 Pa)

V2 = Final volume (unknown, to be determined)

Rearranging the equation to solve for V2:

V2 = (P1 / P2)^(1/γ) * V1

Now, let's plug in the values:

V2 = (400,000 Pa / 2,000,000 Pa)^(1/1.4) * 0.04 m^3

V2 ≈ 0.016 m^3

Now, we can calculate the final temperature (T2) using the ideal gas law:

P2 * V2 = n * R * T2

T2 = (P2 * V2) / (n * R)

T2 = (2,000,000 Pa * 0.016 m^3) / (n * 8.314 J/(mol K))

Next, we need to determine the number of moles (n) of CO2 using the ideal gas law:

P1 * V1 = n * R * T1

n = (P1 * V1) / (R * T1)

n = (400,000 Pa * 0.04 m^3) / (8.314 J/(mol K) * 310 K)

Now, let's calculate T2:

T2 ≈ (2,000,000 Pa * 0.016 m^3) / ((400,000 Pa * 0.04 m^3) / (8.314 J/(mol K) * 310 K))

T2 ≈ 1600 K

Now, we can calculate the change in internal energy (ΔU):

ΔU = (γ - 1) * n * R * (T2 - T1)

ΔU = (1.4 - 1) * [(400,000 Pa * 0.04 m^3) / (8.314 J/(mol K) * 310 K)] * 8.314 J/(mol K) * (1600 K - 310 K)

Finally, we can calculate the minimum work required for the compression process as:

W = -ΔU

Keep in mind that these calculations are based on certain assumptions and ideal gas behavior. For real-world scenarios, additional factors may need to be considered.