ch20 - 1(a Since the gas is ideal its pressure p is given...

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1. (a) Since the gas is ideal, its pressure p is given in terms of the number of moles n , the volume V , and the temperature T by p = nRT / V . The work done by the gas during the isothermal expansion is 2 2 1 1 2 1 ln . = = = ³ ³ V V V V dV V W p dV n RT n RT V V We substitute V 2 = 2.00 V 1 to obtain ( )( )( ) 3 = ln2.00 = 4.00 mol 8.31 J/mol K 400 K ln2.00 = 9.22 10 J. W n RT × (b) Since the expansion is isothermal, the change in entropy is given by ( ) 1 S T dQ Q T = = ³ , where Q is the heat absorbed. According to the first law of thermodynamics, E int = Q W . Now the internal energy of an ideal gas depends only on the temperature and not on the pressure and volume. Since the expansion is isothermal, E int = 0 and Q = W . Thus, 3 9.22 10 J = = = 23.1 J/K. 400 K × W S T (c) S = 0 for all reversible adiabatic processes.
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2. From Eq. 20-2, we obtain ( )( ) 4 = = 405 K 46.0 J/K =1.86 10 J. × Q T S
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3. An isothermal process is one in which T i = T f which implies ln( T f / T i ) = 0. Therefore, with V f / V i = 2.00, Eq. 20-4 leads to ( )( ) ( ) = ln = 2.50 mol 8.31 J/mol K ln 2.00 =14.4 J/K. f i V S nR V § · ¨ ¸ © ¹
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4. (a) This may be considered a reversible process (as well as isothermal), so we use S = Q / T where Q = Lm with L = 333 J/g from Table 19-4. Consequently, S = 333 12.0 273 =14.6 J / g g K J / K. a fa f (b) The situation is similar to that described in part (a), except with L = 2256 J/g, m = 5.00 g, and T = 373 K. We therefore find S = 30.2 J/K.
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5. We use the following relation derived in Sample Problem 20-2: = ln . § · ¨ ¸ © ¹ f i T S mc T (a) The energy absorbed as heat is given by Eq. 19-14. Using Table 19-3, we find ( )( ) 4 J = = 386 2.00 kg 75 K = 5.79 10 J kg K § · × ¨ ¸ © ¹ Q cm T where we have used the fact that a change in Kelvin temperature is equivalent to a change in Celsius degrees. (b) With T f = 373.15 K and T i = 298.15 K, we obtain ( ) J 373.15 = 2.00 kg 386 ln =173 J/K. kg K 298.15 § · § · ¨ ¸ ¨ ¸ © ¹ © ¹ S
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6. An isothermal process is one in which T i = T f which implies ln ( T f / T i ) = 0. Therefore, Eq. 20-4 leads to ( ) ( ) 22.0 = ln = = 2.75 mol. 8.31 ln 3.4/1.3 § · ¡ ¨ ¸ © ¹ f i V S nR n V
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7. (a) The energy that leaves the aluminum as heat has magnitude Q = m a c a ( T ai T f ), where m a is the mass of the aluminum, c a is the specific heat of aluminum, T ai is the initial temperature of the aluminum, and T f is the final temperature of the aluminum- water system. The energy that enters the water as heat has magnitude Q = m w c w ( T f T wi ), where m w is the mass of the water, c w is the specific heat of water, and T wi is the initial temperature of the water. The two energies are the same in magnitude since no energy is lost. Thus, ( ) ( ) + = = . + ¡ a a ai w w wi a a ai f w w f wi f a a w w m c T m c T m c T T m c T T T m c m c The specific heat of aluminum is 900 J/kg K and the specific heat of water is 4190 J/kg K. Thus, ( )( )( ) ( )( )( ) ( )( ) ( )( ) 0.200 kg 900 J/kg K 100 C + 0.0500 kg 4190 J/kg K 20 C = 0.200 kg 900 J/kg K + 0.0500 kg 4190 J/kg K = 57.0 C or 330 K.
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