Physics Solution Manual for 1100 and 2101

# D the speed of the wave is equal to its frequency

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Unformatted text preview: ∆T = Q / Q = C p n ∆T = 5 R n ∆T . Thus, 2 . c Rn h Using these expressions for ∆V and ∆T, we find that – nR ∆T / P – nR Q / bRn g – Q s= = = 5 2 5 2 A = b – –2093 J 5 2 5 2 PA g 1 c.01 × 10 Pa h3.14 × 10 c 5 –2 m2 PA = h 0 . 264 m 94. REASONING According to the conservation of energy, the work W done by the electrical energy is W = QH − QC , where QH is the magnitude of the heat delivered to the outside (the hot reservoir) and QC is the magnitude of the heat removed from the house (the cold reservoir). Dividing both sides of this relation by the time t, we have W t = QH t − QC t The term W / t is the magnitude of the work per second that must be done by the electrical energy, and the terms QH / t and QC / t are, respectively, the magnitude of the heat per second delivered to the outside and removed from the house. Since the air conditioner is a Carnot air conditioner, we know that QH / QC is equal to the ratio TH / TC of the Kelvin temperatures of the hot and cold reservoirs, QH / QC = TH / TC (Equation 15.14). This expression, along with the one above for W / t , will allow us to determine the magnitude of the work per second done by the electrical energy. SOLUTION Solving the expression QH / QC = TH / TC for QH , substituting the result into the relation W t = QH t − QC t , and recognizing that QC / t = 10 500 J/s, give Chapter 15 Problems 825 QC TH W t = QH t − QC t = TC t − QC t Q T 306.15 K = C H − 1 = (10 500 J/s ) − 1 = 5.0 ×102 J/s t TC 292.15 K In this result we have used the fact that TH = 273.15 + 33.0 °C = 306.15 K and TC = 273.15 + 19.0 °C = 292.15 K. Pressure (×105 Pa) 95. REASONING a. The work done by the gas is equal to the area under the pressure-versus-volume curve. We will measure this area by using the graph given with the problem. 6.00 4.00 B A 2.00 0 0 2.00 4.00 6.00 8.00 10.0 12.0 Volume, m3 b. Since the gas is an ideal gas, it obeys the ideal gas law, PV = nRT (Equation 14.1). This implies that PAVA / TA = PBVB / TB . All the variables except for TB in this relation are known. Therefore, we can use this expression to find the temperature at point B. c. The heat Q that has been added to or removed from the gas can be obtained from the first law of thermodynamics, Q = ∆U + W (Equation 15.1), where ∆U is the change in the internal energy of the gas and W is the work done by the gas. The work W is known from part (a) of the problem. The change ∆U in the internal energy of the gas can be obtained from Equation 14.7, ∆U = U B − U A = 3 nR (TB − TA ) , where n is the number of moles, R is 2 the universal gas constant, and TB and TA are the final and initial Kelvin temperatures. We do not know n, but we can use the ideal gas law (PV = nRT ) to replace nRTB by PBVB and to replace nRTA by PAVA. 826 THERMODYNAMICS SOLUTION a. From the drawing we see that the area under the curve is 5.00 “squares,” where each ( )( ) square has an area of 2.00 × 105 Pa 2.00 m3 = 4.00 × 105 J. Therefore, the work W done by the gas is ( ) W = ( 5.00 squares ) 4.00 ×105 J/square = 2.00 × 106 J b. In the Reasoning section, we have seen that PAVA / TA = PBVB / TB . Solving this relation for the temperature TB at point B, using the fact that PA = PB (see the graph), and taking the values for VB and VA from the graph, we have that PV TB = B B P V AA V 10.0 m3 TA = B TA = (185 K ) = 925 K V 2.00 m3 A c. From the Reasoning section we know that the heat Q that has been added to or removed from the gas is given by Q = ∆U + W. The change ∆U in the internal energy of the gas is ∆U = U B − U A = 3 nR (TB − TA ) . Thus, the heat can be expressed as 2 Q = ∆U + W = 3 nR (TB − TA ) + W 2 We now use the ideal gas law (PV = nRT ) to replace nRTB by PBVB and nRTA by PAVA. The result is 3 Q = 2 ( PBVB − PAVA ) + W Taking the values for PB, VB, PA, and VA from the graph and using the result from part a 6 that W = 2.00 × 10 J, we find that the heat is Q= 3 2 ( PBVB − PAVA ) + W ( )( )( )( ) = 3 2.00 ×105 Pa 10.0 m3 − 2.00 ×105 Pa 2.00 m3 + 2.00 ×106 J = 4.40 ×106 J 2 96. REASONING We think of the entire universe as divided into two parts: the sun and the rest of the universe. As the sun spontaneously radiates heat into space, the change in its entropy each second is ∆Ssun , and the change in the entropy of the rest of the universe is ∆Srest each second. The total entropy change of the universe in one second is the sum of the two entropy changes ∆Suniverse = ∆Ssun + ∆Srest (1) Chapter 15 Problems 827 The spontaneous thermal radiation of the sun is an irreversible process, but if we imagine a reversible process by which the sun loses an amount Q of heat, and a reversible process by Q which the rest of the universe absorbs this heat, then we can make use of ∆S = T R (Equation 15.18) to calculate the entropy changes of the sun and of the rest of the universe: ∆Ssun = −Q Tsun ∆S rest = and +Q Trest (2) In Equations (2), Tsun is the sun’s surface temperature, and Tre...
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