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Q. 3.10

Expert-verified
Found in: Page 97

### An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

# An ice cube (mass $30\mathrm{g}$) $0°\mathrm{C}$ is left sitting on the kitchen table, where it gradually melts. The temperature in the kitchen is $25°\mathrm{C}$.(a) Calculate the change in the entropy of the ice cube as it melts into water at $0°\mathrm{C}$. (Don't worry about the fact that the volume changes somewhat.)(b) Calculate the change in the entropy of the water (from the melted ice) as its temperature rises from $0°\mathrm{C}$ to $25°\mathrm{C}$.(c) Calculate the change in the entropy of the kitchen as it gives up heat to the melting ice/water.(d) Calculate the net change in the entropy of the universe during this process. Is the net change positive, negative, or zero? Is this what you would expect?

(a) The required net change in entropy is $\Delta {S}_{1}=36.70{\mathrm{JK}}^{-1}$.

(b) Change in entropy of water is ${S}_{{\mathrm{H}}_{2}\mathrm{O}}=47.70{\mathrm{JK}}^{-1}$.

(c) Change in entropy of kitchen is $\Delta {S}_{\text{room}}=44.14{\mathrm{JK}}^{-1}$.

(d) Change in entropy of universe is $\Delta {S}_{\text{universe}}=3.56{\mathrm{JK}}^{-1}$.

See the step by step solution

## Step 1: Given (part a)

Mass of ice cube, $m=30\mathrm{g}$

Initial temperaturelocalid="1647139485052" $=T=0°\mathrm{C}=273\mathrm{K}$

Temperature in the kitchenlocalid="1647158003634" $=25°C=298\mathrm{K}$

## Step 2: Calculation (part a)

The change in entropy is given as:

$\Delta S=\frac{Q}{T}..........\left(1\right)$

Where,

$Q$ = heat added

role="math" localid="1647139806158" $T$ = Temperature in Kelvin

The amount of heat required to melt ice is given as:

$Q=m×L$

Where,

$m$ = mass

$L$ = Latent heat of water = $334J{g}^{-1}$

By substituting the values in the above equation, we get,

role="math" localid="1647140235964" $Q=30×334\phantom{\rule{0ex}{0ex}}Q=10020J$

Now, by substituting the value of $Q$ in the equation (1), we get,

$\Delta {S}_{1}=\frac{10020}{273}\phantom{\rule{0ex}{0ex}}\Delta {S}_{1}=36.70{\mathrm{JK}}^{-1}$

## Step 3: Final answer (part a)

Hence, the required net change in entropy is calculated as $36.70J{K}^{-1}$.

## Step 4: Given (part b)

Mass of ice cube, $m=30\mathrm{g}$

Initial temperature$=T=0°\mathrm{C}=273\mathrm{K}$

Temperature in the kitchenlocalid="1647158013609" $=25°\mathrm{C}=298\mathrm{K}$

## Step 5: Calculation (part b)

If an amount of heat $Q$ flows from a substance at a constant temperature, the change in entropy is given as:

$\Delta {S}_{2}={C}_{V}{\int }_{{T}_{i}}^{{T}_{f}}\frac{1}{T}dT$

Where,

${C}_{v}$ = Specific heat

$T$ = Temperature in Kelvin

The water absorbs heat as it warms up, but the temperature changes. As a result, it is preferable to employ the entropy-heat-capacity relationship, as well as the fact that the specific heat of water is roughly constant over the liquid range at $c=4.18{\mathrm{Jg}}^{-1}{\mathrm{K}}^{-1}$.

Hence, by further solving the above equation, we get:

$\Delta {S}_{2}={C}_{V}\left[\mathrm{ln}T{\right]}_{{T}_{i}}^{{T}_{f}}\phantom{\rule{0ex}{0ex}}\Delta {S}_{2}={C}_{V}\left[\mathrm{ln}\frac{{T}_{f}}{{T}_{i}}\right]$

But, we know that,

${C}_{V}=mc$

By substituting it, we get,

$\Delta {S}_{2}=mc\left[\mathrm{ln}\frac{{T}_{f}}{{T}_{i}}\right]$

Now substitute the values in the above equation:

$\Delta {S}_{2}=30×4.18×\left[\mathrm{ln}\frac{298}{273}\right]\phantom{\rule{0ex}{0ex}}\Delta {S}_{2}=11J{K}^{-1}$

Thus, the total entropy increase of the water becomes:

${S}_{{H}_{2}O}=\Delta {S}_{1}+\Delta {S}_{2}\phantom{\rule{0ex}{0ex}}{S}_{{H}_{2}O}=36.70+11\phantom{\rule{0ex}{0ex}}{S}_{{H}_{2}O}=47.70J{K}^{-1}$

## Step 6: Final answer (part b)

Hence, the change in entropy of water is calculated as $47.70J{K}^{-1}$.

## Step 7: Given (part c)

Mass of ice cube, $m=30g$

Initial temperature $=T=0°\mathrm{C}=273\mathrm{K}$

Temperature in the kitchen $=25°\mathrm{C}=298\mathrm{K}$

## Step 8: Calculation (part c)

If an amount of heat $Q$ flows into a substance at a constant temperature $T$, the change in entropy is given as:

localid="1647197183247" role="math" $\Delta {S}_{room}=\frac{Q}{T}$

Where,

$Q$ = heat added

$T$ = Temperature in Kelvin

Total amount of heat transferred to the water from the room's air is given as:

$Q={Q}_{ice}+{Q}_{\text{warm}}$

Where,

${Q}_{ice}$ = Heat added from ice

${Q}_{warm}$ = Heat added due to temperature change

We know that,

Total amount of heat transferred to the water from the room's air is calculated as:

$Q={Q}_{ice}+mc\Delta T$

In this case,

As calculated before in part (a), ${Q}_{ice}=10020$ and $c=4.18J{K}^{-1}{g}^{-1}$

Now, by substituting these values and the given values in the above equation, we get,

$Q=\left(10020\right)+\left[4.18×30×\left(298-273\right)\right]\phantom{\rule{0ex}{0ex}}Q=13155J$

Now, by substituting this value, change in entropy can be calculated as:

$\Delta {S}_{\text{room}}=-\frac{Q}{T}\phantom{\rule{0ex}{0ex}}\Delta {S}_{\text{room}}=-\frac{13155}{298}\phantom{\rule{0ex}{0ex}}\Delta {S}_{\text{room}}=-44.14J{K}^{-1}$

## Step 9: Final answer (part c)

Hence, the entropy change of the kitchen is calculated as $-44.14J{K}^{-1}$.

## Step 10: Given (part d)

Mass of ice cube, $m=30g$

Initial temperature $=T=0°\mathrm{C}=273\mathrm{K}$

Temperature in the kitchen $=25°\mathrm{C}=298\mathrm{K}$

Net change of entropy of water role="math" localid="1647197282233" ${S}_{{\mathrm{H}}_{2}\mathrm{O}}=47.70{\mathrm{JK}}^{-1}$

## QStep 11: Calculation (part d)

If an amount of heat $Q$ flows into a substance at a constant temperature $T$, the change in entropy is given as:

$\Delta S=\frac{Q}{T}$

Where,

$Q$ = heat added

$T$ = Temperature in Kelvin

The net change of entropy of the universe for this case can be given as:

$\Delta S=\Delta {S}_{\text{room}}+{S}_{{H}_{2}O}$

Where,

${S}_{room}$ = net change of entropy of the kitchen

${S}_{{H}_{2}O}$ = net change of entropy of water

By substituting the calculated values in the above equation, we get,

$\Delta S=-44.14+47.70\phantom{\rule{0ex}{0ex}}\Delta S=3.56J{K}^{-1}$

## Step 12: Final answer (part d)

Hence, the net change of entropy of the universe is calculated as $3.56J{K}^{-1}$.

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