It is well established that the adiabatic and quasistatic volume change of an id

Question

It is well established that the adiabatic and quasistatic volume change of an ideal gas (R, cv) in a cylinder and piston apparatus follows the path PVk = constant, where k = cp/cv. Real processes, however, depart from this path; the actual path is named polytropic and is represented by the function PVn = constant where n is a constant (n doesn

It is well established that the adiabatic and quasistatic volume change of an ideal gas (R, cv) in a cylinder and piston apparatus follows the path PVk = constant, where k = cp/cv. Real processes, however, depart from this path; the actual path is named polytropic and is represented by the function PVn = constant where n is a constant (n doesn't equal k). One possible reason for this departure is the heat transfer that takes place between the ideal gas and the massive wall of the cylinder. Determine how the constant exponent n is affected by the heat capacity of the cylinder wall. The simplest model that retains the effect of gas-wall heat transfer is presented below. The mass of the idea gas is m, the mass of the cylinder wall is M, and the specific heat of the wall material is c. Consider now the expansion from an initial volume V1 and Pressure P1 to a final volume V2, and assume that at any instant during this process, the ideal gas and the wall material are in mutual thermal equilibrium. The expansion is sufficiently slow so that Delta W = P dV. Furthermore, the combined system (ideal gas and cylinder material) does not exchange heat with its surroundings. Show that the path of the process is PVn = constant, where n = 1 + (R/cv)/(1+Mc/mcv) How large (or how small) should the wall heat capacity be if the path is to approach PVk = constant? Evaluate the entropy change do for the combined system (ideal gas and cylinder material) during the infinitesimal change from V to V + dV.

Explanation / Answer

Consider the wall and the gas together as a system,

Now this system is assumed to be adiabatic,

dq= du+dw

dw=PdV

dq=0 for the system mentioned above...

du= m(Cv)dT+ McdT (both the gas and the wall are heat sinks or sources)

Replace P=mRT/V from ideal gas law...

and setup the differential form,

(m(Cv)+Mc)(dT/T)=-mR(dV/V)

Integrate to obtain

T^(m(Cv)+Mc) *V^mR = constant (After simplifying)

Use ideal gas equation to replace T by PV

and simplify to the form PV^n =const.

u will get

n=1+ (R/(Cv))/(1+ (M/m)*(R/(Cv))

for n--->k

M-->0

So, if M-->0, n=1+(R/(Cv))=(Cv+R)/(Cv)

for ideal gases, Cp=Cv+R

so n-->k=(Cp)/(Cv)

As the whole system is adiabatic,

dq(rev)=0

So, dS=dq/T =0

Plz rate, it took a lot of effort....In case of any doubts, do comment, I will reply asap...

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