Physics Solution Manual for 1100 and 2101

Reasoning according to the kinetic theory of gases

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Unformatted text preview: nner tube, we can apply Boyle’s law: PVi = Pf Vf i (14.3) We will use Equations (1) and Equation 14.3 to determine the distance d = hi − hf through which the pump handle moves before air begins to flow. We note that the final pressure Pf is equal to the pressure of air in the inner tube. SOLUTION Since the cross-sectional area A of the piston does not change, substituting Equations (1) into Equation 14.3 yields Pi A hi = Pf A hf Pi hi = Pf hf or (2) Solving Equation (2) for the unknown final height hf, we obtain hf = Pi hi Pf (3) With Equation (3), we can now calculate the distance d = hi − hf that the biker pushes down on the handle before air begins to flow from the pump to the inner tube: 736 THE IDEAL GAS LAW AND KINETIC THEORY d = hi − hf = hi − P = hi 1 − i P Pf f Phi i 1.0 × 105 Pa = ( 0.55 m ) 1 − 2.4 × 105 Pa = 0.32 m 23. SSM WWW REASONING AND SOLUTION a. Since the heat gained by the gas in one tank is equal to the heat lost by the gas in the other tank, Q1 = Q2, or (letting the subscript 1 correspond to the neon in the left tank, and letting 2 correspond to the neon in the right tank) cm1∆T1 = cm2 ∆T2 , cm1 (T − T1 ) = cm2 (T2 − T ) m1 (T − T1 ) = m2 (T2 − T ) Solving for T gives T= m2T2 + m1T1 m2 + m1 (1) The masses m1 and m2 can be found by first finding the number of moles n1 and n2. From the ideal gas law, PV = nRT, so n1 = ( 5.0 ×105 Pa ) ( 2.0 m3 ) PV1 1 = = 5.5 ×102 mol RT1 [8.31 J/(mol ⋅ K) ] ( 220 K ) This corresponds to a mass m1 = (5.5×10 2 mol) ( 20.179 g/mol ) = 1.1×104 g = 1.1×101 kg . Similarly, n2 = 2.4 × 102 mol and m2 = 4.9 × 103 g = 4.9 kg. Substituting these mass values into Equation (1) yields T= ( 4.9 kg ) ( 580 K ) + (1.1× 101 kg ) ( 220 K ) = ( 4.9 kg ) + (1.1×101 kg ) 3.3 × 102 K b. From the ideal gas law, nRT ( 5.5 ×102 mol ) + ( 2.4 ×102 mol ) [8.31 J/ ( mol ⋅ K )] ( 3.3 ×102 K ) = = 2.8 × 105Pa 3 V 2.0 m + 5.8 m3 ______________________________________________________________________________ P= Chapter 14 Problems 737 24. REASONING Since the temperature of the confined air is constant, Boyle's law applies, and PsurfaceVsurface = PhVh , where P surface and Vsurface are the pressure and volume of the air in the tank when the tank is at the surface of the water, and Ph and Vh are the pressure and volume of the trapped air after the tank has been lowered a distance h below the surface of the water. Since the tank is completely filled with air at the surface, Vsurface is equal to the volume Vtank of the tank. Therefore, the fraction of the tank's volume that is filled with air when the tank is a distance h below the water's surface is Vh Vtank = Vh Vsurface = Psurface Ph We can find the absolute pressure at a depth h using Equation 11.4. Once the absolute pressure is known at a depth h, we can determine the ratio of the pressure at the surface to the pressure at the depth h. SOLUTION According to Equation 11.4, the trapped air pressure at a depth h = 40.0 m is Ph = Psurface + ρ gh = (1.01×105 Pa) + (1.00 ×103 kg/m3 )(9.80 m/s 2 )(40.0 m) = 4.93 × 105 Pa where we have used a value of ρ = 1.00 × 103 kg/m3 for the density of water. The desired volume fraction is 1.01 × 10 5 Pa = 0.205 Vtank Ph 4.93 × 10 5 Pa ______________________________________________________________________________ Vh = Psurface = 25. REASONING The mass (in grams) of the air in the room is the mass of the nitrogen plus the mass of the oxygen. The mass of the nitrogen is the number of moles of nitrogen times the molecular mass (in grams/mol) of nitrogen. The mass of the oxygen can be obtained in a similar way. The number of moles of each species can be found by using the given percentages and the total number of moles. To obtain the total number of moles, we will apply the ideal gas law. If we substitute the Kelvin temperature T, the pressure P, and the volume V (length × width × height) of the room in the ideal gas law, we can obtain the total PV number nTotal of moles of gas as nTotal = , because the ideal gas law does not distinguish RT between types of ideal gases. SOLUTION Using m to denote the mass (in grams), n to denote the number of moles, and M to denote the molecular mass (in grams/mol), we can write the mass of the air in the room as follows: 738 THE IDEAL GAS LAW AND KINETIC THEORY m = mNitrogen + mOxygen = nNitrogen M Nitrogen + nOxygen M Oxygen We can now express this result using f to denote the fraction of a species that is present and nTotal to denote the total number of moles: m = nNitrogen M Nitrogen + nOxygen M Oxygen = f Nitrogen nTotal M Nitrogen + f Oxygen nTotal M Oxygen According to the ideal gas law, we have nTotal = PV . With this substitution, the mass of RT the air becomes ( ) ( m = f Nitrogen M Nitrogen + fOxygen M Oxygen nTotal = f Nitrogen M Nitrogen + f Oxygen M Oxygen ) PV RT The mass per mole for nitrogen (N2) and for oxygen (O2) can be obtained from the periodic table on the inside of the back cover of the text. They are, respectively, 28.0 and 32.0 g/mol. The temperature of 22 ºC must be expressed on the Kelvin scale as 295 K (see Equation 12.1). The mass of the air is, then, ( m = f Nitrogen M Nitrogen + fOxygen M Ox...
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