91Ó°ÊÓ

The number of moles of an ideal gas, n = 0.550 mol

$$\begin{gathered} Forcasea,T=250K \\ Forcasea,V_f=0.8c{m}^3 \\ Forcasea,V_i=0.2c{m}^3 \\ Forcaseb,T=350K \\ Forcaseb,V_f=0.8c{m}^3 \\ Forcaseb,V_i=0.2c{m}^3 \\ Forcasec,T=400K \\ Forcasec,V_f=1.2c{m}^3 \\ Forcasec,V_i=0.3c{m}^3 \\ Forcased,T=450K \\ Forcased,V_f=1.2c{m}^3 \\ Forcased,V_f=0.3c{m}^3 \end{gathered}$$

We use the equation for change in entropy related to work and temperature. The heat for an isothermal process can be expressed in terms of the final and initial volume.

Formulae:

The entropy change of gas,$∆\mathrm{S}=\frac{\mathrm{Q}}{\mathrm{T}}$ (1)

The heat transferred by the gas,$\mathrm{Q}=\mathrm{nRTln}\left(\frac{{\mathrm{V}}_{\mathrm{f}}}{{\mathrm{V}}_{\mathrm{i}}}\right)$ (2)

So, the equation of change in entropy by substituting equation (ii) in equation (i) can be given as:

$$\begin{gathered} ∆\mathrm{S}=\frac{\mathrm{nRTln}\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{f}}}{{\mathrm{V}}_{\mathrm{i}}}}\right)}{\mathrm{T}} \\ =\mathrm{nRTln}\left(\frac{{\mathrm{V}}_{\mathrm{f}}}{{\mathrm{V}}_{\mathrm{i}}}\right) \end{gathered}$$…â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦(3)

Now, let us consider each situation.

Entropy change for part (a) by using the given values in equation (3) as given:

$$\begin{gathered} ∆\mathrm{S}=0.550\mathrm{mol}×8.314\mathrm{J}/\mathrm{mol}.\mathrm{K}×\ln \left(\frac{0.8{\mathrm{m}}^3}{0.2{\mathrm{m}}^3}\right) \\ =6.34\mathrm{J}/\mathrm{K} \end{gathered}$$

Entropy change for part (b) by using the given values in equation (3) as given:

$$\begin{gathered} ∆\mathrm{S}=0.550\mathrm{mol}×8.314\mathrm{J}/\mathrm{mol}.\mathrm{K}×\ln \left(\frac{0.8{\mathrm{m}}^3}{0.2{\mathrm{m}}^3}\right) \\ =6.34\mathrm{J}/\mathrm{K} \end{gathered}$$

Entropy change for part (c) by using the given values in equation (3) as given:

$$\begin{gathered} ∆\mathrm{S}=0.550\mathrm{mol}×8.314\mathrm{J}/\mathrm{mol}.\mathrm{K}×\ln \left(\frac{1.2{\mathrm{m}}^3}{0.2{\mathrm{m}}^3}\right) \\ =6.34\mathrm{J}/\mathrm{K} \end{gathered}$$

Entropy change for part d by using the given values in equation (3) as given:

$$\begin{gathered} ∆\mathrm{S}=0.550\mathrm{mol}×8.314\mathrm{J}/\mathrm{mol}.\mathrm{K}×\ln \left(\frac{1.2{\mathrm{m}}^3}{0.2{\mathrm{m}}^3}\right) \\ =6.34\mathrm{J}/\mathrm{K} \end{gathered}$$

Hence, the values of the entropy change for all the cases (a), (b), (c), and (d) are found to be 6.34 J/K in each case.