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Chapter 20 Heat and the First Law of Thermodynamics

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Mechanical Equivalent of Heat Joule found that it took approximately 4.18 J of mechanical energy to raise the temperature of 1g water at 1 o C Joule found that it took approximately 4.18 J of mechanical energy to raise the temperature of 1g water at 1 o C Later, more precise measurements determined the amount of mechanical energy needed to raise the temperature of 1g of water from 14.5 o C to 15.5 o C Later, more precise measurements determined the amount of mechanical energy needed to raise the temperature of 1g of water from 14.5 o C to 15.5 o C 1 cal = J 1 cal = J This is known as the mechanical equivalent of heat

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Heat Capacity The heat capacity, C, of a particular sample is defined as the amount of energy needed to raise the temperature of that sample by 1 o CThe heat capacity, C, of a particular sample is defined as the amount of energy needed to raise the temperature of that sample by 1 o C If energy Q produces a change of temperature of T, thenIf energy Q produces a change of temperature of T, then Q = C T

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Specific Heat Specific heat, c, is the heat capacity per unit massSpecific heat, c, is the heat capacity per unit mass If energy Q transfers to a sample of a substance of mass m and the temperature changes by T, then the specific heat isIf energy Q transfers to a sample of a substance of mass m and the temperature changes by T, then the specific heat is

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Specific Heat The specific heat is essentially a measure of how insensitive a substance is to the addition of energyThe specific heat is essentially a measure of how insensitive a substance is to the addition of energy –The greater the substance’s specific heat, the more energy that must be added to cause a particular temperature change The equation is often written in terms of Q :The equation is often written in terms of Q : Q = m c T Q = m c T

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Some Specific Heat Values

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More Specific Heat Values

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Sign Conventions If the temperature increases:If the temperature increases: Q and T are positive Energy transfers into the system If the temperature decreases:If the temperature decreases: Q and T are negative Energy transfers out of the system

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Specific Heat Varies With Temperature Technically, the specific heat varies with temperature Technically, the specific heat varies with temperature The corrected equation is The corrected equation is However, if the temperature intervals are not too large, the variation can be ignored and c can be treated as a constant However, if the temperature intervals are not too large, the variation can be ignored and c can be treated as a constant There is only about a 1% specific heat variation between 0 o and 100 o C

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Specific Heat of Water Water has the highest specific heat of common materialsWater has the highest specific heat of common materials This is responsible for many weather phenomenaThis is responsible for many weather phenomena –Moderate temperatures near large bodies of water –Global wind systems –Land and sea breezes

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Calorimetry One technique for measuring specific heat involves heating a material, adding it to a sample of water, and recording the final temperatureOne technique for measuring specific heat involves heating a material, adding it to a sample of water, and recording the final temperature This technique is known as calorimetryThis technique is known as calorimetry –A calorimeter is a device in which this energy transfer takes place

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Calorimetry The system of the sample and the water are isolatedThe system of the sample and the water are isolated Conservation of energy requires that the amount of energy that leaves the sample equals the amount of energy that enters the waterConservation of energy requires that the amount of energy that leaves the sample equals the amount of energy that enters the water –Conservation of Energy gives a mathematical expression of this: Q cold = -Q hot

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Calorimetry The negative sign in the equation is critical for consistency with the established sign convention The negative sign in the equation is critical for consistency with the established sign convention Since each Q = mc T, c sample can be found by: Since each Q = mc T, c sample can be found by: Technically, the mass of the container should be included, but if m w >>m container it can be neglected

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Phase Changes A phase change is happened when a substance changes from one form to anotherA phase change is happened when a substance changes from one form to another –Two common phase changes are Solid to liquid (melting)Solid to liquid (melting) Liquid to gas (boiling)Liquid to gas (boiling) During a phase change, there is no change in temperature of the substanceDuring a phase change, there is no change in temperature of the substance

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Latent Heat Different substances react differently to the energy added or removed during a phase change due to their different molecular arrangements Different substances react differently to the energy added or removed during a phase change due to their different molecular arrangements The amount of energy also depends on the mass of the sample The amount of energy also depends on the mass of the sample

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Latent Heat If an amount of energy Q is required to change the phase of a sample of mass m, If an amount of energy Q is required to change the phase of a sample of mass m, L = Q /m The quantity L is called the latent heat of the material The quantity L is called the latent heat of the material Latent means “hidden” The value of L depends on the substance as well as the actual phase change The energy required to change the phase is The energy required to change the phase is Q = mL

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Latent Heat The latent heat of fusion is used when the phase change is from solid to liquidThe latent heat of fusion is used when the phase change is from solid to liquid The latent heat of vaporization is used when the phase change is from liquid to gasThe latent heat of vaporization is used when the phase change is from liquid to gas The positive sign is used when the energy is transferred into the systemThe positive sign is used when the energy is transferred into the system –This will result in melting or boiling The negative sign is used when energy is transferred out of the systemThe negative sign is used when energy is transferred out of the system –This will result in freezing or condensation

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Sample Latent Heat Values

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Graph of Ice to Steam

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Warming Ice, Graph Part A Start with one gram of ice at –30.0ºC Start with one gram of ice at –30.0ºC During phase A, the temperature of the ice changes from –30.0ºC to 0ºC. The specific heat of ice is 2090 J/kg· 0 C During phase A, the temperature of the ice changes from –30.0ºC to 0ºC. The specific heat of ice is 2090 J/kg· 0 C Q = m i c i ΔT = (1x10 -3 kg)(2090 J/kg· 0 C)( C)=62.7J In this case, 62.7 J of energy are added

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Melting Ice, Graph Part B Once at 0ºC, the phase change (melting) starts Once at 0ºC, the phase change (melting) starts The temperature stays the same although energy is still being added. The latent heat of fusion for water is 3.33x10 5 J/kg The temperature stays the same although energy is still being added. The latent heat of fusion for water is 3.33x10 5 J/kg Q =m i L f =(1x10 -3 kg)(3.33x10 5 J/kg)=333J The energy required is 333 J On the graph, the values move from 62.7 J to 396 J

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Warming Water, Graph Part C Between 0ºC and 100ºC, the material is liquid and no phase changes take place Between 0ºC and 100ºC, the material is liquid and no phase changes take place Energy added increases the temperature. Specific heat for water 4186 J/kg· 0 C. Energy added increases the temperature. Specific heat for water 4186 J/kg· 0 C. Q = m w c w ΔT = 419J. 419 J are added The total is now 815 J

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Boiling Water, Graph Part D At 100ºC, a phase change occurs (boiling) At 100ºC, a phase change occurs (boiling) Temperature does not change. The latent heat of vaporization for water is 2.26x10 6 J/kg. Temperature does not change. The latent heat of vaporization for water is 2.26x10 6 J/kg. Use Q = m w L v. This requires 2260 J Use Q = m w L v. This requires 2260 J The total is now 3075 J

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Heating Steam After all the water is converted to steam, the steam will heat up After all the water is converted to steam, the steam will heat up No phase change occurs No phase change occurs The added energy goes to increasing the temperature. The specific heat for steam is 2010J/kg∙ 0 C. The added energy goes to increasing the temperature. The specific heat for steam is 2010J/kg∙ 0 C. Use Use Q = m s c s ΔT In this case, 40.2 J are needed The temperature is going to 120 o C The total is now 3115 J The total is now 3115 J

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Molecular View of Phase Changes Phase changes can be described in terms of the rearrangement of molecules (or atoms in an elemental substance) Phase changes can be described in terms of the rearrangement of molecules (or atoms in an elemental substance) Liquid to Gas phase change Liquid to Gas phase change Molecules in a liquid are close together The forces between them are stronger than those in a gas Work must be done to separate the molecules The latent heat of vaporization is the energy per unit mass needed to accomplish this separation

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Molecular View of Phase Changes Solid to Liquid phase change The addition of energy will cause the amplitude of the vibration of the molecules about their equilibrium position to increase The addition of energy will cause the amplitude of the vibration of the molecules about their equilibrium position to increase At the melting point, the amplitude is great enough to break apart bonds between the molecules At the melting point, the amplitude is great enough to break apart bonds between the molecules The molecules can move to new positions The molecules can move to new positions The molecules in the liquid are bound together less strongly than those of the solid The molecules in the liquid are bound together less strongly than those of the solid The latent heat of fusion is the energy per unit mass required to go from the solid-type to the liquid-type bonds The latent heat of fusion is the energy per unit mass required to go from the solid-type to the liquid-type bonds

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Molecular View of Phase Changes The latent heat of vaporization is greater than the latent heat of fusion The latent heat of vaporization is greater than the latent heat of fusion In the liquid-to-gas phase change, the liquid- type bonds are broken The gas molecules are essentially not bonded to each other It takes more energy to completely break the bonds than to change the type of bonds It takes more energy to completely break the bonds than to change the type of bonds

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Calorimetry Problem-Solving Strategy Units of measure must be consistent Units of measure must be consistent For example, if your value of c is in J/kg.o C, then your mass must be in kg, the temperatures in o C and energies in J Transfers of energy are given by Transfers of energy are given by Q =mc T only when no phase change occurs If there is a phase change, use Q = mL If there is a phase change, use Q = mL

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Calorimetry Problem-Solving Strategy Be sure to select the correct sign for all energy transfersBe sure to select the correct sign for all energy transfers Remember to use Q cold = - Q hotRemember to use Q cold = - Q hot –The T is always T f - T i

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If water with a mass m h at temperature T h is poured into an aluminum cup of mass m Al containing mass m c of water at T c, where T h > T c, what is the equilibrium temperature of the system?

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State Variables State variables describe the state of a system State variables describe the state of a system In the macroscopic approach to thermodynamics, variables are used to describe the state of the system In the macroscopic approach to thermodynamics, variables are used to describe the state of the system Pressure, temperature, volume, internal energy These are examples of state variables The macroscopic state of an isolated system can be specified only if the system is in thermal equilibrium internally The macroscopic state of an isolated system can be specified only if the system is in thermal equilibrium internally

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Work in Thermodynamics Work can be done on a deformable system, such as a gas Work can be done on a deformable system, such as a gas Consider a cylinder with a moveable piston Consider a cylinder with a moveable piston A force is applied to slowly compress the gas A force is applied to slowly compress the gas The compression is slow enough for all the system to remain essentially in thermal equilibrium This is said to occur quasi- statically

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Work The piston is pushed downward by a force F through a displacement of dr: The piston is pushed downward by a force F through a displacement of dr: A·dy is the change in volume of the gas, dV Therefore, the work done on the gas is dW = -P dV dW = -P dV

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Work Interpreting dW = - P dVInterpreting dW = - P dV –If the gas is compressed, dV is negative and the work done on the gas is positive –If the gas expands, dV is positive and the work done on the gas is negative –If the volume remains constant, the work done is zero The total work done is:The total work done is:

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PV Diagrams The state of the gas at each step can be plotted on a graph called a PV diagram This allows us to visualize the process through which the gas is progressing The curve is called the path The curve is called the path PV diagrams can be used when the pressure and volume are known at each step of the process PV diagrams can be used when the pressure and volume are known at each step of the process

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PV Diagrams The work done on a gas in a quasi- static process that takes the gas from an initial state to a final state is the negative of the area under the curve on the PV diagram, evaluated between the initial and final states The work done on a gas in a quasi- static process that takes the gas from an initial state to a final state is the negative of the area under the curve on the PV diagram, evaluated between the initial and final states – This is true whether or not the pressure stays constant – The work done does depend on the path taken

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Work Done By Various Paths Each of these processes has the same initial and final statesEach of these processes has the same initial and final states The work done differs in each processThe work done differs in each process The work done depends on the pathThe work done depends on the path

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Work From a PV Diagram The volume of the gas is first reduced from V i to V f at constant pressure P i The volume of the gas is first reduced from V i to V f at constant pressure P i Next, the pressure increases from P i to P f by heating at constant volume V f Next, the pressure increases from P i to P f by heating at constant volume V f W = -P i (V f – V i ) W = -P i (V f – V i )

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Work From a PV Diagram The pressure of the gas is increased from P i to P f at a constant volume The pressure of the gas is increased from P i to P f at a constant volume The volume is decreased from V i to V f The volume is decreased from V i to V f W = -P f (V f – V i ) W = -P f (V f – V i )

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Work From a PV Diagram The pressure and the volume continually change The pressure and the volume continually change The work is some intermediate value between P f (V f – V i ) and P i (V f – V i ) The work is some intermediate value between P f (V f – V i ) and P i (V f – V i ) To evaluate the actual amount of work, the function P(V) must be known To evaluate the actual amount of work, the function P(V) must be known V

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A sample of ideal gas is expanded to twice its original volume of 1.00 m 3 in a quasi-static process for which P = V 2, with = 5.00 atm/m 6, as shown in Figure. How much work is done on the expanding gas?

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A sample of ideal gas is expanded to twice its original volume of 1.00 m 3 in a quasi-static process for which P = V 2, with = 5.00 atm/m 6, as shown in Figure. How much work is done on the expanding gas? The work done on the gas is the negative of the area under the curve between and.

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A sample of ideal gas is expanded to twice its original volume of 1.00 m 3 in a quasi-static process for which P = V 2, with = 5.00 atm/m 6, as shown in Figure. How much work is done on the expanding gas?.

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(a) Determine the work done on a fluid that expands from i to f as indicated in Figure. (b) How much work is performed on the fluid if it is compressed from f to i along the same path?

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(a)(b)

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Heat Transfer The energy transfer, Q, into or out of a system also depends on the process The energy transfer, Q, into or out of a system also depends on the process The piston is pushed upward, the gas is doing work on the piston The piston is pushed upward, the gas is doing work on the piston

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Heat Transfer This gas has the same initial volume, temperature and pressure as the previous example This gas has the same initial volume, temperature and pressure as the previous example The final states are also identical The final states are also identical No energy is transferred by heat through the insulating wall No energy is transferred by heat through the insulating wall No work is done by the gas expanding into the vacuum No work is done by the gas expanding into the vacuum

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Energy Transfer, Summary Energy transfers by heat, like the work done, depend on the initial, final, and intermediate states of the systemEnergy transfers by heat, like the work done, depend on the initial, final, and intermediate states of the system Both work and heat depend on the path takenBoth work and heat depend on the path taken Neither can be determined solely by the end points of a thermodynamic processNeither can be determined solely by the end points of a thermodynamic process

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The First Law of Thermodynamics The First Law of Thermodynamics is a special case of the Law of Conservation of Energy The First Law of Thermodynamics is a special case of the Law of Conservation of Energy It takes into account changes in internal energy and energy transfers by heat and work Although Q and W each are dependent on the path, Q + W is independent of the path Although Q and W each are dependent on the path, Q + W is independent of the path The First Law of Thermodynamics states that E int = Q + W The First Law of Thermodynamics states that E int = Q + W All quantities must have the same units of measure of energy

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The First Law of Thermodynamics One consequence of the first law is that there must exist some quantity known as internal energy which is determined by the state of the system One consequence of the first law is that there must exist some quantity known as internal energy which is determined by the state of the system For infinitesimal changes in a system dE int = dQ + dW For infinitesimal changes in a system dE int = dQ + dW The first law is an energy conservation statement specifying that the only type of energy that changes in a system is internal energy The first law is an energy conservation statement specifying that the only type of energy that changes in a system is internal energy

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Isolated Systems An isolated system is one that does not interact with its surroundings An isolated system is one that does not interact with its surroundings No energy transfer by heat takes place No energy transfer by heat takes place The work done on the system is zero The work done on the system is zero Q = W = 0, so E int = 0 The internal energy of an isolated system remains constant The internal energy of an isolated system remains constant

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Cyclic Processes A cyclic process is one that starts and ends in the same state A cyclic process is one that starts and ends in the same state – This process would not be isolated – On a PV diagram, a cyclic process appears as a closed curve The change in internal energy must be zero since it is a state variable The change in internal energy must be zero since it is a state variable If E int = 0, Q = -W In a cyclic process, the net work done on the system per cycle equals the area enclosed by the path representing the process on a PV diagram In a cyclic process, the net work done on the system per cycle equals the area enclosed by the path representing the process on a PV diagram

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Adiabatic Process An adiabatic process is one during which no energy enters or leaves the system by heat An adiabatic process is one during which no energy enters or leaves the system by heat Q = 0 This is achieved by: –Thermally insulating the walls of the system –Having the process proceed so quickly that no heat can be exchanged

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Adiabatic Process Since Q = 0, E int = WSince Q = 0, E int = W If the gas is compressed adiabatically, W is positive so E int is positive and the temperature of the gas increasesIf the gas is compressed adiabatically, W is positive so E int is positive and the temperature of the gas increases If the gas expands adiabatically, the temperature of the gas decreasesIf the gas expands adiabatically, the temperature of the gas decreases

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Adiabatic Processes Some important examples of adiabatic processes related to engineering are: Some important examples of adiabatic processes related to engineering are: − The expansion of hot gases in an internal combustion engine − The liquefaction of gases in a cooling system − The compression stroke in a diesel engine

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Adiabatic Free Expansion This is an example of adiabatic free expansion This is an example of adiabatic free expansion The process is adiabatic because it takes place in an insulated container The process is adiabatic because it takes place in an insulated container Because the gas expands into a vacuum, it does not apply a force on the membrane and W = 0 Because the gas expands into a vacuum, it does not apply a force on the membrane and W = 0 Since Q = 0 and W = 0, E int = 0 and the initial and final states are the same Since Q = 0 and W = 0, E int = 0 and the initial and final states are the same – No change in temperature is expected

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Isobaric Processes An isobaric process is one that occurs at a constant pressure An isobaric process is one that occurs at a constant pressure The values of the heat and the work are generally both nonzero The values of the heat and the work are generally both nonzero The work done isW = P (V f – V i )where is the constant pressure The work done is W = P (V f – V i ) where P is the constant pressure

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Isovolumetric Processes An isovolumetric process is one in which there is no change in the volume An isovolumetric process is one in which there is no change in the volume Since the volume does not change, W = 0 Since the volume does not change, W = 0 From the first law, E int = Q From the first law, E int = Q If energy is added by heat to a system kept at constant volume, all of the transferred energy remains in the system as an increase in its internal energy If energy is added by heat to a system kept at constant volume, all of the transferred energy remains in the system as an increase in its internal energy

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Isothermal Process An isothermal process is one that occurs at a constant temperature An isothermal process is one that occurs at a constant temperature Since there is no change in temperature, E int = 0 Since there is no change in temperature, E int = 0 Therefore, Q = - W Therefore, Q = - W Any energy that enters the system by heat must leave the system by work Any energy that enters the system by heat must leave the system by work

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Isothermal Process At right is a PV diagram of an isothermal expansion At right is a PV diagram of an isothermal expansion The curve is a hyperbola The curve is a hyperbola The curve is called as isotherm The curve is called as isotherm

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Isothermal Expansion The curve of the PV diagram indicates PV = constant The curve of the PV diagram indicates PV = constant The equation of a hyperbola Because it is an ideal gas and the process is quasi-static, PV = nRT and Because it is an ideal gas and the process is quasi-static, PV = nRT and

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Isothermal Expansion Numerically, the work equals the area under the PV curve Numerically, the work equals the area under the PV curve The shaded area in the diagram If the gas expands, V f > V i and the work done on the gas is negative If the gas expands, V f > V i and the work done on the gas is negative If the gas is compressed, V f < V i and the work done on the gas is positive If the gas is compressed, V f < V i and the work done on the gas is positive

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Special Processes, Summary Adiabatic Adiabatic No heat exchanged Q = 0 and E int = W Isobaric Isobaric Constant pressure W = P (V f – V i ) and E int = Q + W Isothermal Isothermal Constant temperature E int = 0 and Q = -W

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A sample of an ideal gas goes through the process shown in Figure. From A to B, the process is adiabatic; from B to C, it is isobaric with 100 kJ of energy entering the system by heat. From C to D, the process is isothermal; from D to A, it is isobaric with 150 kJ of energy leaving the system by heat. Determine the difference in internal energy E int,B – E int,A.

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T=constant

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Mechanisms of Heat Transfer We want to know the rate at which energy is transferred There are various mechanisms responsible for the transfer: –Conduction –Convection –Radiation

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Conduction The transfer can be viewed on an atomic scale The transfer can be viewed on an atomic scale –It is an exchange of energy between microscopic particles by collisions The microscopic particles can be atoms, molecules or free electrons –Less energetic particles gain energy during collisions with more energetic particles Rate of conduction depends upon the characteristics of the substance Rate of conduction depends upon the characteristics of the substance

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Conduction example The molecules vibrate about their equilibrium positions The molecules vibrate about their equilibrium positions Particles near the heat source vibrate with larger amplitudes Particles near the heat source vibrate with larger amplitudes These collide with adjacent molecules and transfer some energy These collide with adjacent molecules and transfer some energy Eventually, the energy travels entirely through the pan Eventually, the energy travels entirely through the pan

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Conduction In general, metals are good conductors In general, metals are good conductors They contain large numbers of electrons that are relatively free to move through the metal They can transport energy from one region to another Poor conductors include asbestos, paper, and gases Poor conductors include asbestos, paper, and gases Conduction can occur only if there is a difference in temperature between two parts of the conducting medium Conduction can occur only if there is a difference in temperature between two parts of the conducting medium

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Conduction The slab at right allows energy to transfer from the region of higher temperature to the region of lower temperature The slab at right allows energy to transfer from the region of higher temperature to the region of lower temperature The rate of energy transfer is given by: The rate of energy transfer is given by:

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Conduction A is the cross-sectional area A is the cross-sectional area Δx is the thickness of the slab Δx is the thickness of the slab or the length of a rod or the length of a rod P is in Watts when Q is in Joules and t is in seconds P is in Watts when Q is in Joules and t is in seconds k is the thermal conductivity of the material k is the thermal conductivity of the material Good conductors have high k values and good insulators have low k values

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Temperature Gradient The quantity |dT / dx| is called the temperature gradient of the material The quantity |dT / dx| is called the temperature gradient of the material It measures the rate at which temperature varies with position For a rod, the temperature gradient can be expressed as: For a rod, the temperature gradient can be expressed as:

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Rate of Energy Transfer in a Rod Using the temperature gradient for the rod, the rate of energy transfer becomes:

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Compound Slab For a compound slab containing several materials of various thicknesses (L 1, L 2, …) and various thermal conductivities (k 1, k 2, …) the rate of energy transfer depends on the materials and the temperatures at the outer edges:For a compound slab containing several materials of various thicknesses (L 1, L 2, …) and various thermal conductivities (k 1, k 2, …) the rate of energy transfer depends on the materials and the temperatures at the outer edges:

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Some Thermal Conductivities

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More Thermal Conductivities

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In Figure, the change in internal energy of a gas that is taken from A to C is +800 J. The work done on the gas along path ABC is –500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the cycle goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is +500 J, how much energy must be added to the system by heat as it goes from point C to point D?

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In Figure, the change in internal energy of a gas that is taken from A to C is +800 J. The work done on the gas along path ABC is –500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the cycle goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is +500 J, how much energy must be added to the system by heat as it goes from point C to point D? (conservation of energy) (a)(First Law)

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In Figure, the change in internal energy of a gas that is taken from A to C is +800 J. The work done on the gas along path ABC is –500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the cycle goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is +500 J, how much energy must be added to the system by heat as it goes from point C to point D? (b),, and Then, (+ means that work is done on the system)

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In Figure, the change in internal energy of a gas that is taken from A to C is +800 J. The work done on the gas along path ABC is –500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the cycle goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is +500 J, how much energy must be added to the system by heat as it goes from point C to point D? (c) so that (– means that energy must be removed from the system by heat)

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In Figure, the change in internal energy of a gas that is taken from A to C is +800 J. The work done on the gas along path ABC is –500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the cycle goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is +500 J, how much energy must be added to the system by heat as it goes from point C to point D? (d) and

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A bar of gold is in thermal contact with a bar of silver of the same length and area. One end of the compound bar is maintained at 80.0°C while the opposite end is at 30.0°C. When the energy transfer reaches steady state, what is the temperature at the junction?

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In the steady state condition, so that In this case

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A bar of gold is in thermal contact with a bar of silver of the same length and area. One end of the compound bar is maintained at 80.0°C while the opposite end is at 30.0°C. When the energy transfer reaches steady state, what is the temperature at the junction? In this case where T is the temperature of the junction. Therefore,

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