Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

In an experiment, 200 g of aluminum (with a specific heat of 900J/kg.K) at 100 $°$ C is mixed with 50.0g of water at 20.0 $°$ C, with the mixture thermally isolated.(a)What is the equilibrium temperature?(b)What is the entropy changes of the aluminum, (c)What is the entropy changes of the water, and (d)What is the entropy changes of the aluminum – water system?

a) The equilibrium temperature of aluminum-water system is 330K.

b) Change in entropy of the aluminum is $- 22.1 J / K$ .

c) Change in entropy of the water is 24.9 J/K

d) Change in entropy of the aluminum-water system is 2.8 J/K

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

a) Mass of the water, $m_{w} = 50.0 g or 50 \times 10^{- 3} kg$

b) Mass of the aluminum, $m_{a l} = 200 g o r 200 \times 10^{- 3} kg$

c) Temperature of aluminum, $T_{a l} = 100 ° C or 373 K$

d) Temperature of water, $T_{w} = 20.0 ° C or 293 K$

e) Specific heat of aluminum, $c_{a l} = 900 J / kg . K$

f) Specific heat of water, $c_{a l} = 4190 J / kg . K$

Step 2: Understanding the concept of equilibrium and entropy change

Entropy change is a phenomenon that quantifies how disorder or randomness has changed in a thermodynamic system. We can write the condition of equilibrium for heat energy absorbed and transferred by the two blocks, and by writing it in terms of specific heat and temperature; we can find the value of equilibrium temperature of the two-block system. Then, by using the formula for change in entropy, we can find the change in entropy of the aluminum, water, and aluminum-water system.

Formulae:

The heat transferred by the body, $Q = m c ∆ T$ …(i)

The entropy change of the gas, $∆ S = m c l n (\frac{T_{f}}{T_{i}})$ …(ii)

Step 3: (a) Calculation of the equilibrium temperature of aluminum-water system

At equilibrium, the total heat of the system is zero. So, $Q_{w} + Q_{a l} = 0$

Using equation (i) and the given values, we can get that

role="math" localid="1661323263437" $m_{w} c_{w} ∆ T_{w} + m_{a l} c_{a l} ∆ T_{a l} = 0$

Let the equilibrium temperature be $T_{f}$ Then, the above equation becomes

$m_{w} c_{w} (T_{f} - T_{w}) + m_{a l} c_{a l} (T_{f} - T_{a l} = 0 50 \times 10^{- 3} kg (4190 J / kg . K) (T_{f} - 293 K) + 200 \times 10^{- 3} kg (900 J / kg . K) (T_{f} - 373 K) = 0 209.5 T_{f} - 61383.5 + 180 T_{f} - 67140 = 128523.5 389.5 T_{f} = 128523.5 T_{f} = 329.97 K \approx 330 K$

Hence, the value of the equilibrium temperature is 330K

Step 4: (b) Calculation of the entropy change of aluminum

Substituting the given values in equation (ii), we get the entropy change of aluminum as given:

$∆ S = 200 \times 10^{- 3} kg (900 J / kg . K) I n (\frac{329.97 K}{373 K}) = - 22.06 J / K = - 22.1 J / K$

Hence, the value of the entropy change is $- 22.1 J / K$

Step 5: (c) Calculation of the entropy change of water

Substituting the given values in equation (ii), we get the entropy change of water as given:

$∆ S = 50 \times 10^{- 3} kg (4190 J / kgK) I n (\frac{329.97 K}{293 K}) = 24.89 J / K \approx 24.9 J / K$

Hence, the value of the entropy change is 24.9 J/K

Step 6: (d) Calculation of the entropy change of aluminum-water system

Change in entropy of the system = Change in entropy of water + Change in entropy of aluminum.

So, the total entropy change of the aluminum-water system is given as follows:

$∆ S = ∆ S_{w} + ∆ S_{a l} = - 22.06 J / K + 24.89 J / K \approx 2.8 J / K$

Hence, the value of the entropy change of the system is 2.8J/K

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of equilibrium and entropy change

Step 3: (a) Calculation of the equilibrium temperature of aluminum-water system

Step 4: (b) Calculation of the entropy change of aluminum

Step 5: (c) Calculation of the entropy change of water

Step 6: (d) Calculation of the entropy change of aluminum-water system

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of equilibrium and entropy change

Step 3: (a) Calculation of the equilibrium temperature of aluminum-water system

Step 4: (b) Calculation of the entropy change of aluminum

Step 5: (c) Calculation of the entropy change of water

Step 6: (d) Calculation of the entropy change of aluminum-water system

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