Isothermal Compression of an Ideal Gas in a Frictionless PCD

[Nah346]

A frictionless piston-cylinder compresses 1 mol of nitrogen to a tenth of its original volume isothermally at 300 K. Assume nitrogen is an ideal gas with Cp*=(7/2)R.

(a) Show two different approaches to calculate the compression work required.
(b) What is the corresponding heat flow?
(c) If the piston-cylinder now experiences significant friction, how will the heat and work flows be different in order to accomplish the same thermodynamic change?
(d) Can the two different approaches in (a) be used to calculate the compression work for a piston cylinder with significant friction? Explain.

 

[Navin A S]

(a) Two different approaches to calculate the compression work required:1. Using an energy balance approach: The work done is equal to the change in internal energy, since no heat is exchanged with the surroundings. The change in internal energy of the nitrogen is U2-U1=Cv(T2-T1)= (5/2)R(T2-T1). Since T2=T1=300K, the work done is zero.2. Using a specific volume-pressure relationship: The work done is equal to the integral of PdV. The pressure-volume relationship for an ideal gas is PV=nRT, so the work done is W=∫PdV=∫nRT/VdV=nRTln(V2/V1)=nRTln(1/10). Substituting n=1 mol, R=8.314 J/mol K, and V1=10V2, the work done is W=-83.14 J.(b) The corresponding heat flow is zero, since the process is isothermal.(c) If the piston-cylinder experiences significant friction, the heat and work flows will be different in order to accomplish the same thermodynamic change. The work done will increase due to the additional friction, while the heat flow will increase as well to provide the energy necessary to overcome the friction.(d) The two different approaches in (a) cannot be used to calculate the compression work for a piston cylinder with significant friction, since they assume no friction. A more complex approach would be needed to take into account the effect of friction on the heat and work flows.