Edit to “Solution”
en/5.9.3.md
+2 −2
| ### Statement | |||
| $5.9.3.$ How much will the entropy of $1 \text{ kg}$ of water at a temperature of $293 \text{ K}$ increase when it is converted to steam? | |||
| ### Solution | |||
| Resulting change of entropy can be represented as the sum of changes of the entropy in each stage: heating, then evaporation. | |||
| $$ | |||
| \Delta S = \Delta S_{1} + \Delta S_{2} | |||
| $$ | |||
| @@ -13,7 +13,7 @@Solution | |||
| I. Heating the water: As $\delta Q=cmdT$: | |||
| $$ | |||
| − | \Delta S_{1} =\int_{T_{1}}^{T_{2}}\frac{\delta Q}{T} = cm\int_{T_{1}}^{T_{2}}\frac{dT}{T} = cm\cdot ln\frac{T_{2}}{T_{1}} | ||
| + | \Delta S_{1} =\int_{T_{1}}^{T_{2}}\frac{\delta Q}{T} = cm\int_{T_{1}}^{T_{2}}\frac{dT}{T} = cm\cdot \ln\frac{T_{2}}{T_{1}} | ||
| $$ | |||
| Where $T_{1} = 293~\text{K}$ and $T_{2} = 373~\text{K}$ and $c = 4.2~\mathrm{\frac{kJ}{kg\cdot \text{ K}}}$ is the specific heat of water. | |||
| II. When water evaporates the temperature is constant. | |||
| $$ | |||
| \Delta S_{2} = \int_{1}^{2}\frac{\delta Q}{T} = \frac{1}{T_{2}}\int_{1}^{2}\delta Q = \frac{Q}{T_{2}} = \frac{mL}{T_{2}} | |||
| $$ | |||
| Here $L$ is the heat of vaporization of the water. | |||
| @@ -29,7 +29,7 @@Solution | |||
| The resulting change in entropy: | |||
| $$ | |||
| − | \boxed{\Delta S =cm\cdot ln\frac{T_{2}}{T_{1}} + \frac{mL}{T_{2}}} | ||
| + | \boxed{\Delta S =cm\cdot \ln\frac{T_{2}}{T_{1}} + \frac{mL}{T_{2}}} | ||
| $$ | |||
| $$ | |||
| \Delta S = 7189 ~\mathrm{\frac{J}{K}} \approx 7.2 ~\mathrm{\frac{kJ}{K}} | |||
| $$ | |||
| #### Answer | |||
| $$ | |||
| \Delta S = 7.2 ~\mathrm{\frac{kJ}{K}} | |||
| $$ | |||
| unchanged lines 8 | |||
| ### Statement | ### Statement | ||
| $5.9.3.$ How much will the entropy of $1 \text{ kg}$ of water at a temperature of $293 \text{ K}$ increase when it is converted to steam? | $5.9.3.$ How much will the entropy of $1 \text{ kg}$ of water at a temperature of $293 \text{ K}$ increase when it is converted to steam? | ||
| ### Solution | ### Solution | ||
| Resulting change of entropy can be represented as the sum of changes of the entropy in each stage: heating, then evaporation. | Resulting change of entropy can be represented as the sum of changes of the entropy in each stage: heating, then evaporation. | ||
| $$ | $$ | ||
| \Delta S = \Delta S_{1} + \Delta S_{2} | \Delta S = \Delta S_{1} + \Delta S_{2} | ||
| $$ | $$ | ||
| @@ -13,7 +13,7 @@Solution | |||
| I. Heating the water: As $\delta Q=cmdT$: | I. Heating the water: As $\delta Q=cmdT$: | ||
| $$ | $$ | ||
| \Delta S_{1} =\int_{T_{1}}^{T_{2}}\frac{\delta Q}{T} = cm\int_{T_{1}}^{T_{2}}\frac{dT}{T} = cm\cdot ln\frac{T_{2}}{T_{1}} | \Delta S_{1} =\int_{T_{1}}^{T_{2}}\frac{\delta Q}{T} = cm\int_{T_{1}}^{T_{2}}\frac{dT}{T} = cm\cdot \ln\frac{T_{2}}{T_{1}} | ||
| $$ | $$ | ||
| Where $T_{1} = 293~\text{K}$ and $T_{2} = 373~\text{K}$ and $c = 4.2~\mathrm{\frac{kJ}{kg\cdot \text{ K}}}$ is the specific heat of water. | Where $T_{1} = 293~\text{K}$ and $T_{2} = 373~\text{K}$ and $c = 4.2~\mathrm{\frac{kJ}{kg\cdot \text{ K}}}$ is the specific heat of water. | ||
| II. When water evaporates the temperature is constant. | II. When water evaporates the temperature is constant. | ||
| $$ | $$ | ||
| \Delta S_{2} = \int_{1}^{2}\frac{\delta Q}{T} = \frac{1}{T_{2}}\int_{1}^{2}\delta Q = \frac{Q}{T_{2}} = \frac{mL}{T_{2}} | \Delta S_{2} = \int_{1}^{2}\frac{\delta Q}{T} = \frac{1}{T_{2}}\int_{1}^{2}\delta Q = \frac{Q}{T_{2}} = \frac{mL}{T_{2}} | ||
| $$ | $$ | ||
| Here $L$ is the heat of vaporization of the water. | Here $L$ is the heat of vaporization of the water. | ||
| @@ -29,7 +29,7 @@Solution | |||
| The resulting change in entropy: | The resulting change in entropy: | ||
| $$ | $$ | ||
| \boxed{\Delta S =cm\cdot ln\frac{T_{2}}{T_{1}} + \frac{mL}{T_{2}}} | \boxed{\Delta S =cm\cdot \ln\frac{T_{2}}{T_{1}} + \frac{mL}{T_{2}}} | ||
| $$ | $$ | ||
| $$ | $$ | ||
| \Delta S = 7189 ~\mathrm{\frac{J}{K}} \approx 7.2 ~\mathrm{\frac{kJ}{K}} | \Delta S = 7189 ~\mathrm{\frac{J}{K}} \approx 7.2 ~\mathrm{\frac{kJ}{K}} | ||
| $$ | $$ | ||
| #### Answer | #### Answer | ||
| $$ | $$ | ||
| \Delta S = 7.2 ~\mathrm{\frac{kJ}{K}} | \Delta S = 7.2 ~\mathrm{\frac{kJ}{K}} | ||
| $$ | $$ | ||
| unchanged lines 8 | |||