Consider n moles of a gas, initially confined within a volume V and held at temp

Question

Consider n moles of a gas, initially confined within a volume V and held at temperature T. the gas

please have legible handwriting. and please show why certain steps where used.

this is a thermal physics problem

is expanded to a total volume oVv constant, by (a) a reversible isothermal and (b) removing a partition and alowiba expansion into the vacuum. Both c trated in Fig. 14.9. Assuming theare derive an expression for the change of the gas in each case. the gas is ent rop partitio / (a-1)V (a-1)V gns Fig. 14.9 Diagram showing n moles of gas, ini confined within a volume V Repeat this calculation for case (a), assumin the gas obeys the van der Waals equation of stat (p+ n a /2 )(V-nb) = nRT (14SE Show further that for case (b) the temperature the van der Waals gas falls by an amount prop tional to (a -1)/a.

Explanation / Answer

For deriving this we will start with Tds equation i.e.
TdS = dU+ PdV where T is the temperature, dS is the change in entropy, dU is the change in internal energy , P is pressure and dV is the infinetly small volume
now
dS= (du/T) + (PdV/T)    now from vanderwall equation we will replace the P/T in the terms of volume
and dU = mcdT since change in the temp is zero therefore dU will be zero , hence
dS = (P/T)dV now we put the value and on integration we will get the value.
For the value of P/T we will divide the vanderwall equation by T and put that value in the above equation.
we get heat or TdS equal to this hence calculate the value of dS.

While free expansion against vaccum the workdone will be zero, and assuming it as thermal insulating
we follow the above process and then
which can be used in numerical purpose as

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