Thermodynamics Problem - Adiabatic Reversible Process....

[Allison Barry]

Homework Statement
A specific type of ideal gas has a specific heat capacity at constant pressure (cp=cv+R) that is a function of temperature T, such that cp=0.5+876T, where cp has units of J/kg/K and T has units of K. The gas, which is initially at T1 = 294 K and P1 = 1x105 Pa, undergoes a reversible adiabatic process such that its final temperature is T2 = 778 K. Calculate the pressure of the gas (in Pa) in this final state. Assume the following ideal gas constant: R = 287 J/kg/K. Recall that ds = cpdT/T – RdP/P.

Relevant equations
Reversible Adiabatic process:
Entropy is 0 because this is adiabatic and reversible.
Constituitive relation for entropy of such a process is:
S₂ - S₁ = ∫₁²dS = 0
dS = c_p(dT/T) - R(dP/P) = 0
Attempt:
Integrating I got:
∫_T1^T2(0.5 + 876T).(dT/T) - 287∫_P1^P2(dP/P)
876(778 - 294) + (0.5) ln (778/294) = 287ln(P₂ / 100,000Pa)
Getting 1488.810788 = lnP₂

But e^1488.810788 is undefinable? I am confused? Did I integrate this wrong?

 

[Chestermiller]

That 876 looks very, very, very suspicious. Are you sure it's not 0.0876 or 0.00876 or, more likely, 0.000876? There's definitely something wrong here.

Your methodology and arithmetic are correct, however.

Chet

 

[Allison Barry]

That 876 looks very, very, very suspicious. Are you sure it's not 0.0876 or 0.00876 or, more likely, 0.000876? There's definitely something wrong here.

Your methodology and arithmetic are correct, however.

Chet

876 is part of "a +bt", the linear relationship between temperature and C_p, it's 0.5 + 876T

 

[Chestermiller]

876 is part of "a +bt", the linear relationship between temperature and C_p, it's 0.5 + 876T

What I'm saying is that there is something wrong with that value. It can't be correct.