We can approach this problem by using the adiabatic compression equations and the relationship between work, pressure, and volume for reversible and irreversible processes. The adiabatic compression equations are:
p1V1^γ = p2V2^γ
T1V1^(γ-1) = T2V2^(γ-1)
where p is pressure, V is volume, T is temperature, and γ is the specific heat ratio of the gas (which is approximately 1.4 for carbon dioxide).
a) For an ideal gas with a constant-pressure heat capacity of 37.151 J/(mol K), the work required for a reversible adiabatic compression is:
W = -nCp(ΔT) = -nCp(T2 - T1)
where n is the number of moles of gas, Cp is the constant-pressure heat capacity, and ΔT is the change in temperature. Using the adiabatic compression equations, we can solve for the outlet temperature (T2) and the work required for the irreversible compression (W_irr):
p1V1^γ = p2V2^γ
T1V1^(γ-1) = T2V2^(γ-1)
Substituting p1 = 1 bar, p2 = 10 bar, V1 = nRT1/p1, and solving for V2, we get:
V2 = V1(p1/p2)^(1/γ) = (nRT1/p1)(p1/p2)^(1/γ)
Using the second adiabatic compression equation, we can solve for T2:
T2 = T1(p2/p1)^((γ-1)/γ) = T1(10/1)^0.4 = 316 K
Substituting n = 1 mole, Cp = 37.151 J/(mol K), T1 = 25°C = 298 K, and T2 = 316 K into the reversible work equation, we get:
W_rev = -nCp(T2 - T1) = -37.151 J/(mol K) * (316 K - 298 K) = -667.6 J
The work required for the irreversible compression is 25% greater than the reversible work, so we have:
W_irr = 1.25 * W_rev = -834.5 J
b) For an ideal gas with the constant-pressure heat capacity given in Appendix A.II, we can use the same approach as above, but substitute the appropriate Cp value. Let's assume that the Cp value given in Appendix A.II is valid for temperatures between 25°C and 316 K. Then, we have:
Cp = 29.07 J/(mol K)
Substituting this value into the reversible work equation, we get:
W_rev = -nCp(T2 - T1) = -29.07 J/(mol K) * (316 K - 298 K) = -524.3 J
Using the same adiabatic compression equations as above, we can solve for T2 and W_irr:
V2 = (nRT1/p1)(p1/p2)^(1/γ) = (1 mol * 8.314 J/(mol K) * 298 K / 1 bar) * (1 bar / 10 bar)^(1/1.4) = 0.0159 m^3
T2 = T1(p2/p1)^((γ-1)/γ) = 298 K * (10/1)^0.4 = 316 K
W_irr = 1.25 * W_rev = -655.4 J
Therefore, for an ideal gas with the constant-pressure heat capacity given in Appendix A.II, the outlet temperature of the carbon dioxide is 316 K, and the work that must be supplied to the compressor for the reversible and irreversible compressors are -524.3 J and -655.4 J, respectively.