Question

Question #7b328

Answer 1

Well, you haven't specified the conditions (what temperature? What pressure?), but you also haven't said anything about how the temperature changes (or doesn't change??)... I guess I'll have to assume constant temperature... shame on me.

`DeltaS_"isothermal"(T,V) = C_Vcancel(ln|T_2/T_1|)^(0) + nRln|V_2/V_1|`

I get `"1.44 J/K"` as a result, but you'll have to supply temperatures if you want a nonisothermal calculation.

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Write the total differential of the entropy as a function of temperature `T` and volume `V`:

`dS = ((delS)/(delT))_VdT + ((delS)/(delV))_TdV` `" "bb((1))`

Once we find out what these derivative terms mean, we can integrate `dS` from the initial to final state to find `DeltaS`.

The first derivative term is:

`((delS)/(delT))_V = 1/T ((delU)/(delT))_V = C_V/T`, `" "bb((2))`

where `C_V` is the constant-volume heat capacity and `U` is the internal energy.

The second term is gotten from the Maxwell Relation for the Helmholtz energy as a function of the natural variables `T` and `V`:

`dA = -SdT - PdV`

The cross-derivatives are equal because the Helmholtz free energy is a state function:

`((delS)/(delV))_T = ((delP)/(delT))_V` `" "bb((3))`

Plugging `(2)` and `(3)` into `(1)`, this results in:

`dS = C_V/TdT + ((delP)/(delT))_VdV`

Now, we can't just leave a derivative term here. Evaluating it first would make this easier. Using the ideal gas law, `PV = nRT`, and assuming helium is ideal:

`((delP)/(delT))_V = del/(delT)[(nRT)/V]_V = (nR)/V` `" "bb((4))`

Plugging `(4)` into `(1)`, this means:

`dS = C_V/TdT + nR cdot 1/VdV` `" "bb((5))`

And finally, integrating `(5)` over the initial and final states gives the change in entropy for this process, which `(i)` heats/cools the gas at constant volume and then `(ii)` expands/contracts the gas at constant temperature.

`color(green)(DeltaS(T,V)) = int_((1))^((2)) dS = int_(T_1)^(T_2) C_V/TdT + nR int_(V_1)^(V_2) 1/VdV`

`= color(green)(C_Vln|T_2/T_1| + nRln|V_2/V_1|)`

Evidently, we cannot do this without knowing temperature values... So we assume you are somehow implying constant temperature conditions all the way through...

`=> color(blue)(DeltaS_"isothermal") = C_Vcancel(ln|T_2/T_1|)^(0) + nRln|(2cancel(V_1))/cancel(V_1)|`

`= (1 cancel"g He" xx cancel"1 mol"/(4.0026 cancel"g"))("8.314472 J/"cancel"mol"cdot"K")ln2`

`=` `color(blue)("1.44 J/K")`