PHYSICAL CHEMISTRY I
PHYSICAL CHEMISTRY I CHEM 481
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Date Created: 10/25/15
CHAPTER 3 PROBLEMS THE SECOND LAW OF THERMODYNAMICS Calculation of Entropy Changes Book Problems Exercises 15 Problems 1 3 B 3 Consider 1 mole of a monatomic ideal gas at PA VA TA Calculate ASSVS Assnquot and ASTm if the gas is expanded i adiabatically and reversibly to V2 its initial pressure or ii adiabatically and irreversibly against a constant pressure of V2 its initial pressure Two metal gas cylinders are connected by a valve that is initially closed One cylinder is filled with gas A and has volume VA and the other is filled with gas B and has volume V5 The pressure inside each cylinder is the same ie PAPBP The valve is opened and the gases are allowed to mix Calculate ASS Assnquot and ASTM Substance A nA moles and substance B n5 moles are brought into thermal contact in an insulated container Both A and B are solids and their molar heat capacities CW are on and 3 for A and B respectively a What is the final common temperature b Show that the approach to a common temperature is a spontaneous process c What is A5 for the system and the surroundings One mole of a monatomic perfect gas is simultaneously heated and expanded to a volume that is 10 times the initial volume and a temperature 5 times the initial temperature The initial state is denoted byA PA VA TA and the final state is denoted by B PBVBTB a Draw a l PV diagram depicting a quottwostep path connecting these two states in which the first step is an isothermal reversible expansion to the final volume Label the intermediate state llC b What physical process occurs during the second step c What is PC in terms of PA VA TA d Calculate A5 of the system for the first step e Calculate A5 of the surroundings for the first step Suppose the expansion in the first step was against a constant pressure equal to PC What would be A5 for the system and surroundings during the first step g Calculate A5 of the system for the second step h Calculate AA for the first step f Thermochemistrv Book Problems Exercises 710 5 Calculate the standard Helmholtz energy of formation AA of CH30H at 298 K from the standard Gibbs energy of formation AfG 16627 kJmole Assume that all gases are ideal Fundamental Eguations Book Problems 6 Two quantities that we have already encountered are 0U0VT and 0H0PT In earlier chapters we stated without proof the following equations of state l 3 r Ul le Pl P 6V T 67 V T6V V 6P T 6T P Start with UUVS and HPS respectively and use Maxwell s relationships to derive these two equations of state Use these expressions to show that the HP and UV for an ideal gas are both zero Explain why this makes sense 11 E The volume of a pure substance has the following dependence on pressure P 2 V nVO where V0 is the volume when the pressure is P0 a Determine AG for an increase in pressure from P0 to 10 b What is the chemical potential of this substance as a function of pressure Consider the change in entropy A5 for the isothermal expansions of both an ideal gas and a Van der Waals gas Which is greater Why does this make sense 1 B 3 U1 CHAPTER 1 Problems REAL AND IDEAL GASES Consider a gas with the following equation of state PV B RT where 3 is a positive constant a Show that the compression factor for this gas can be expressed as a function of the molar volume as follows 2 z 1 B Vm Vi b Sketch Z for this gas as a function of Vm and comment on the relative importance of the attractive and repulsive forces c Provide an expression for the work done during the isothermal reversible expansion of this gas from VA to VB d Is this gas capable of doing more or less work than an ideal gas Explain your answer Sketch a single diagram containing the isotherms of a real gas with TgtTc TTc and TltTc and indicate on the diagram the location of the critical point What is the significance of the critical temperature Consider 1 mole of C02 at 50 atm of pressure at 25 0C a Calculate the volume in L occupied by the gas using the ideal gas law b Calculate the volume occupied by the gase in L using the van der Waals equation For COZ a3640 atm L2 mole 2 and b 4267 x 10 2 L mole l c Do attractive or repulsive forces dominate Explain you answer One mole of Ar gas at 273 K is placed in vessel with variable volume As the size of the vessel is decreased at what volume in liters do the repulsive forces between Ar atoms begin to dominate The virial coefficients for Ar at 273 K are BT 00217 Lmole and CT 000120 LZmolez Consider two containers at of gas connected to each other through a tube with a valve Initially the container on the right 5 L volume is filled with 5 moles of A molecules and the container on the left 1 L volume is filled with 1 mole of B molecules The temperature is 300 K When the valve is opened and the gases are allowed to mix A and B react according to 2A B 9 C Since the reaction is exothermic the temperature rises and by the time the reaction is complete the final temperature is 350 K What are the partial pressures in atm of A B and C after the reaction has gone to completion Another equation of state used to describe reagases is the Berthelot equation P VRTb T32 m m a This equation is very similar to the Van der Waals equation Based on what you know about the Van der Waals equation what is the physical origin of the 2quotd term in the Berthelot equation ie the aTVm2 term b These two equations Bethelot and Van der Waals have different mathematical forms Provide a physical explanation for the origin of T in the second term and why T appears in the denominator as opposed to the numerator For the diagram below do the following P latml T165 K V Litermole Calculate the volume for the point marked A assuming that the gas is ideal Sketch on the graph P vs V assuming that the gas is ideal at this temperature Your sketch should pass through A Sketch on the graph P vs V assuming that the gas is real and provide physical explanations for the deviations from ideal behavior One of the most widely used real gas equations of state is the van der Waals equation RT 1 P a 2 Vm b Vm The b coefficient in this equation reflects the quotexcluded volume ie the volume that the molecules themselves take up Hence it is not surprising that this coefficient scales with atomic volume ie b k VAlom NA where NA is Avagadro s number VAmm is the volume of a single atom calculated from its atomic radius and k is a proportionality constant It is easy to show using simple geometrical arguments that for a van der Waals gas k 4 and should be independent of the type of gas One can test the validity of this expression for actual gases eg Ar Kr or Xe by calculating the value for k from two experimentaly known quantities i the b coefficient and ii the atomic radius which gives VAmm The results for a series of noble gases are summarized below 3 G as b NAVAtom kbNAVAtom Ne 00167 00054 308 Ar 00320 00132 240 Kr 00386 00170 226 Xe 00516 00261 197 None of the gases have a proportionality constant equal to 4 but all are fairly close within a factor of 2 There is however a clear trend evident in the table namely k gets progressively smaller as the gas size gets bigger Provide a physical explanation for this trend Could a gas with the following equation of state condense into a liquid at low temperature Use the concept and mathematical expression of the compression factor to explain your answer Vm b 31 3 CHAPTER 2 PROBLEMS THE FIRST LAW OF THERMODYNAMICS Why is the heat capacity of a monatomic ideal gas independent of temperature A sample consisting of4 moles of He is isothermally and reversibly compressed at 0 C from 10 L to 5 L Calculate q w AU and AH for this process Consider one mole of a perfect monatomic gas in state A Le PA VA TA that is reversibly compressed to a final pressure that is 10 times its initial pressure ie P nal1OPA The compression is done two different ways 1 isothermally and 2 adiabatically a Draw a single diagram indicating the isothermal and adiabatic paths Label the final states for the isothermal and adiabatic expansions B and C respectively b Calculate VB Express your answer in terms of VA c show that vC 11035vA d How much heat would have to be added or removed from the gas compressed isothermally state B to bring it to state C Express your answer in terms of PA VA TA andor R e Show that AH for the adiabatic expansion is CPAT One mole of a perfect monatomic gas at a pressure P1 volume V1 and temperature T1 undergoes the following isothermal compressionexpansion cycle First the gas is isothermally and reversibly compressed to a volume that is 13 its original volume Then the gas is heated a constant pressure to four times its initial temperature Express your answers in terms of P1 V1 and T1 and R a Draw a diagram that shows these two paths and label them clearly Label the initial point A the intermediate point B and the final point C Label the volumes and pressures for each point on the xaxis and yaxis in terms of V1 and P1 b Suppose the gas was compressed adiabatically to 13 its initial volume Show this path on the diagram and label the final point D You do not have to calculate the final pressure c Calculate w and q for Step 1 d Calculate AH and AU for Step 2 Consider one mole of a perfect diatomic gas in state A ie PA VA TA that is reversibly compressed to a volume that is one third its initial volume a Draw a single diagram indicating the paths that the system takes if the gas is compressed 1 isothermally or 2 adiabatically Label the final states for the isothermal and adiabatic expansions states B and C respectively b Calculate PB Express your answer in terms of PA VA TA andor R c Show that PC375PA d Determine an expression for TC Express your answer in terms of PA VA TA andor R e Calculate AU for the adiabatic path Consider Ar treated as a real gas described by the following equation of state where b is positive constant p zi V b Show that the expansion coefficient for Ar treated as an ideal gas is lT Derive an expression for the expansion coefficient for the real Ar gas given above 3 bl I Write down an expression for the AH and AU for the following reaction at 298 K NH3g HClg NH4Cls Express your answers in terms of the heats of formation for the reactants and products 3 Consider the combustion of octane Cngg a Write down the balanced chemical equation for this reaction b Calculate AH at 298 K c Calculate AU at 298 K to Show that the following function has an exact differential 5 4 f x y x y 10 Write an expression for the total differential dP given that the pressure is a function of V and T Express your answer in terms of the expansion coefficient and the isothermal compressibility KT 1V0V0PT 11 Consider a real gas described by the following equation of state P 1 mj V V V2 a What equation of state is this b Derive an expression for the isothermal compressibility When using the above equation of state truncate the series to include only the first two terms 12 Consider one mole of a van der Waals gas that has no repulsive interactions This gas is expanded isothermally and reversibly from V1 to 10VL at a temperature T a Derive expressions for W q and AU b Write down an expression for W for an ideal gas Compare this with the results obtained in part a and provide a physical explanation for the difference 13 Consider a liquid whose pressuree can be expressed as a function ofT and V ie PPTV a Write down an expression for the total differential of P ie dP in terms of the expansion coefficient 01 and compressibility K If both the expansion coefficient 01 and compressibility K are constant determine an equation of state for this system ie determine PTV E 14 Consider cylinderpiston arrangement containing 1 mole of gas at some initial temperature T1 and pressure P1 The cylinder is heated at constant pressure a Show that w RAT b For a fixed amount of heat say 3 Joules would system be capable of doing more work if it were filled with Ar or N2 or would the work be the same Explain your answer by calculating the work for each case Rationalize your answer from part b That is provide a physical explanation as to why one is more than the other or both are the same 0 15 Consider a box with an adiabatic boundary that has a small hole The box is filled with an ideal gas The gas is heated and its temperature rises Use the ideal gas law to show that AUO 16 Write down an expression for the internal energy differential dU and use it to show that 6U CV TETOLV 6T P 17 Derive the following expression BCV T azp av 6T2 T V