PHSI191 Module 3 - Thermodynamics (Notes) PDF

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PHSI191 Module 3ThermodynamicsLecture 1Temperature & Thermal EnergyTemperatureTemperature is a measure how hot or cold something is. It is measured in Celsius or Kelvin. Remember that:Thermal EquilibriumFirst, we must take a look at the concept of thermal equilibrium. If two systems, A and B, ar...

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PHSI191 Module 3 Thermodynamics

Lecture 1 Temperature & Thermal Energy Temperature Temperature is a measure how hot or cold something is. It is measured in Celsius or Kelvin. Remember that:

0K = − 273.15∘C

Thermal Equilibrium First, we must take a look at the concept of thermal equilibrium. If two systems, A and B, are in thermal equilibrium, and a third system, C, is in thermal equilibrium with A, then it is also in thermal equilibrium with B. The property that the systems share is called temperature. When we use a thermometer, we place it in contact with an object and allow it to reach thermal equilibrium so it has the same temperature as the object. We can therefore say, for all practical purposes, that temperature is what a thermometer reads. The atoms and molecules of matter are in constant motion. Any gas, liquid, or solid has an amount of kinetic energy associated with this random motion. All of this energy together is the thermal energy. The thermal energy of an object depends on the number of molecules in the object and the molecular composition, as well as the temperature. At higher temperatures, the randomly moving atoms and molecules of an object move faster and the thermal energy is higher. When two objects at different temperatures are placed in contact, collisions occur between the molecules in the two objects. Thermal energy is transferred in this process, and this transferred energy is called heat. The movement of this thermal energy due to a temperature difference is called heat transfer. When the objects in thermal contact are at the same temperature, they are said to be in thermal equilibrium. This is a dynamic equilibrium because the thermal energy exchange process does not stop when equilibrium is reached. At the interface between the two objects there is equal quantities of thermal energy flowing in each direction.

Thermal Expansion Because atoms move faster as temperature is increased, they tend to move further apart. Hence, objects normally expand as temperature increases. For a solid rod of some material:

ΔL = L 0 αΔT ΔL is the change in length of an object, L0 is the original length, and α is the linear coefficient for thermal expansion. The expansion coefficient is a measure of the fractional change in length per degree of temperature change. When a flat plane of some material is heated, it expands in both directions at once. The change in area will be:

ΔA = A 0(2α)ΔT Similarly, for isotropic materials, the change in volume can be calculated by:

ΔV = V0(3α)ΔT

Note that we can have temperature values in either degrees Celsius or Kelvin here, since it is only the change in temperature that is needed.

Lecture 2 Ideal Gases The Gas Laws There are a variety of gas laws we need to understand. Charles’ Law When a fixed quantity of gas is held at a constant pressure, it is found experimentally that the volume of the gas increases linearly with temperature:

V = aT V is the volume of the gas sample, T is the temperature in Kelvin, and a is the proportionality constant. However, a is not dependant on the chemical structure of the gas. This equation is equally valid for any gas, but it fails if the temperature is too low or the density is too high. Boyle’s Law When a fixed quantity of gas is held at a fixed temperature, it is found experimentally that the pressure is inversely proportional to the volume. For example, doubling the pressure will halve the volume of the sample. This is expressed as:

P=

b V

P is the absolute pressure and V is the volume. The proportionality constant, b, depends on the absolute temperature of the gas sample and the number of molecules of gas present, but it does not depend on the type of gas. Ideal Gas Law We can combine these two equations into one, to form:

PV = cT Where c is a proportionality constant. Experimentally, it is found the c depends only on the number of gas molecules present, provided the temperature is not too low and the density is not too high. It does not depend on the mass of the individual atoms or molecules, or their structure. Because c is proportional to the number of molecules, we can write:

c = Nk Where k is a proportionality constant called Boltzmann’s constant (1.381 x 10-23 J K-1). If we substitute this into the equation, we get:

PV = NkT The word “ideal” in ideal gas law is used because the equation holds strictly only in the ideal limit of a gas of very low density and sufficiently high temperature. The ideal gas law can also be written in terms of the number of moles of a gas:

PV = n RT

Where n is the number of moles and R is the universal gas constant (8.314 J K-1 mol-1). Dalton’s Law All the gas laws mentioned so far apply equally well to all gases. This means they can also be applied to mixtures of gases, such as air. In this situation it is useful to use the concept of partial pressure, which refers to the pressure exerted by one individual component of the gas. The partial pressure of that gas is the pressure it would have if only that component of the gas were present in the same volume. The sum of partial pressures of all the component gases is the same as the total pressure of the gas mixture:

Ptotal = P1 + P2 + P3 + . . . If we are interested in only one component of the mixture, molecules of gas of type j, then the pressure in the ideal gas law is the partial pressure of gas j:

Pj =

Nj kT V

Kinetic Theory Of Gases Temperature, measured in Kelvin, provides a direct measure of the average kinetic energy per molecule or atom. More specifically, the average kinetic energy per molecule in a gas (monoatomic gas) is:

1 3 mv 2 = kT 2 2 Multiplying this by the number of molecules or atoms (N), the total kinetic energy of all the molecules in the gas (thermal energy) is:

3 NkT 2 3 = n RT 2

U=

The average speed of molecules is defined by:

vm =

3kT m

However, the molecules of a gas actually have a very wide range of speeds, this is only an average.

Lecture 3 Real Gases Real Gases For real gases, the ideal gas equation fails when the temperature is low enough and the volume is small enough. Then the attractive forces between the atoms of the gas cause the atoms to stick together. The gas then condenses to a liquid or solid. This is called a phase change.

Phase Changes In the region where liquid and vapour co-exist, the pressure of the isotherms (horizontal lines in a PV diagram) is constant. The pressure at these points is uniquely determined by the temperature. This uniquely identified temperature is called the saturated vapour pressure. A phase diagram illustrates how the saturated vapour pressure depends on temperature for each phase transition.

When substances change phase, thermal energy is required to overcome the inter-atomic forces. We have seen that when the phase changes, the temperature remains constant. The thermal energy required for 1 kg of a substance to melt or vaporise is called the latent heat of phase change (L). For a mass of a substance:

Q = mL Q is the thermal energy input required to change the phase of the substance. Latent heat of phase change depends on the conditions, such as temperature. For example, changing ice to water at 0°C has a lower latent heat than changing water to steam at 100°C. When moisture evaporates from the skin, it uses a relatively large amount of thermal energy. This process is used by the body for regulating its temperature. The sign of Q can tell us the direction of phase change: - Q is positive, energy is added to a substance, so either solid to liquid or liquid to gas. - Q is negative, energy is removed from a substance, so either liquid to solid or gas to liquid.

Heat Transfer Without Phase Change The thermal energy transferred to a cold object from a hot one when they are in contact is called heat (Q). Q is positive when the thermal energy of the relevant object increases. If the heat received by the cold object is Q, the corresponding quantity of heat for the hot one will be -Q. The specific heat of a substance is the amount of heat supplied to 1 kg of the substance in order to increase its temperature by 1K or 1°C. It is also called the specific heat capacity. If the temperature of a mass with specific heat increases by a temperature, the heat input is:

Q = m cΔT For an isolate system, when two objects or more objects at different temperatures are brought together and come to thermal equilibrium, the total energy is conserved.

Q1 + Q2 + . . . = 0 We can use energy conservation to determine the final equilibrium temperature of the two objects, utilising the equations above.

Lecture 4 Water Vapour In Air Dew Point Temperature The human body has millions of sweat glands, and it is the evaporation of this sweat which is used for controlling body temperature. We know that this process makes use of the latent heat of water (energy required to vaporise). But this evaporation can be impeded if the existing vapour pressure of water in the atmosphere is high. The partial pressure of water vapour in the air is normally less than the saturated vapour pressure of water at air temperature. Instead, the partial pressure for water vapour in the atmosphere equals the saturated vapour pressure of water at a lower temperature. How do we find this temperature? We must cool down a surface until we get water condensation (dew) forming on the surface. The temperature at which this occurs is called the dew point temperature.

Moisture Content Of Air The moisture content of air is the mass (mw) of moisture per kg of the rest of the air (dry air). This is also called the absolute humidity or the humidity ratio. We can obtain this quantity from the ideal gas law:

m w Mw Pw = ma Ma Pa Where mw and ma is the mass of water and dry air, Mw and Ma are the molar mass of water (18 g mol-1) and air (28.97 g mol-1), and Pw and Pa are the partial pressures of water and dry air. Remember that the sum of partial pressures equals the total pressure, therefore:

Pw + Pa = Pt Pw mw = 0.621 ma Pt − Pw

Psychrometric Chart Now that we know about dew point temperature and the moisture content of air, we can look at a psychrometric chart, which is like a comparison of the two. It is a composite graph to show how the different properties of moist air are related. Normal air temperature is called dry-bulb temperature to distinguish it from dew point temperature. Dew point temperature can be measured by cooling a surface until dew condenses on it. For our purposes, we assume air is at normal atmospheric pressure (101.3 kPa). We know the dew point temperature is less than the temperature of the air because the actual vapour pressure is less than the maximum achievable at air temperature. The relative humidity is defined as the actual partial vapour pressure of water, over the saturated vapour pressure at the same temperature. The psychrometric chart shows lines of different relative humidity using the air temperature and the moisture content of the air as coordinates. For an example, what if we wanted to find the relative humidity, knowing the dew point

temperature was 20°C and the dry-bulb temperature was 35°C? First we find 20°C on the dry-bulb temperature axis. Go all the way up to 100% relative humidity (which is where the dew forms). Then, trace horizontally along the dotted lines until you reach 35°C on the dry-bulb temperature axis. It is here that you will see around what relative humidity you are in.

Lecture 5 Heat Transfer Wet-Bulb Temperature When air passes over a wet surface, moisture evaporates when the relative humidity is less than 100%. As the air gains water vapour, its temperature drops because the air provides heat to the liquid water in order to evaporate it. The process of evaporation and cooling continues until the relative humidity reaches 100%. The temperature is then the wet-bulb temperature. The wet surface can be cooled right down to the wet-bulb temperature but no further. So the wet-bulb temperature is the temperature that a wet sleeve covering a thermometer bulb will reach when air is blown over it. When the relative humidity is low, the wet-bulb temperature will be much less than the dry-bulb temperature. But at 100% relative humidity, the wet-bulb temperature equals the dry-bulb temperature.

Heat Transfer Heat transfer takes place by three processes: - Conduction - takes place in solids, liquids, and gases, where molecules exchange thermal energy without changing their mean position. This is a slow process. - Convection - takes place in liquids and gases, molecules carry thermal energy while changing their mean position. This is a more rapid process. - Radiation - no medium is required, electromagnetic radiation. For an isolated system, the total thermal energy is conserved when two objects are put in thermal contact. Assume that no other object is involved in heat transfer to the combined system, therefore:

Q1 + Q2 = 0 The direction of heat flow is always from hot to cold. Conduction Heat conduction occurs when the moving electrons and molecules exchange energy with other electrons and molecules through collisions. Materials vary in their ability to transfer heat. Good electrical conductors such as metals, have loose outer electrons and so are good heat conductors. Good electrical insulators have no loose electrons, so they are normally good thermal insulators (like wood, glass, etc). The rate of heat transfer for conduction is:

ΔQ k AΔT = d Δt = hcond AΔT Where k is the thermal conductivity of the slab, ΔT is the difference in temperature between the two surfaces, and hcond is the conduction heat transfer coefficient. Convection Convection is like conduction, but at least one of the media involved is a moving fluid. Consider a specific example of a solid and fluid. By continually having new fluid at one side of the interface, heat transfer can be maintained indefinitely. If there is no fluid motion, then the temperature difference would gradually decrease at the

interface, thus lowering the driving force for heat transfer. The rate of heat transfer depends on the velocity of the fluid and many other fluids. For a fluid moving past a stationary surface, the rate of heat transfer to the solid can be estimated as:

ΔQ = hconv AΔT Δt Radiation All objects radiate and absorb energy in the form of electromagnetic waves. This is radiative heat transfer. No contact or medium is required. The rate of transfer due to radiation for heat radiated from one surface is:

ΔQ = AσεT 4 Δt For the neat heat radiated between two surfaces:

ΔQ = hra d AΔT Δt Here, A is the surface area, T is the temperature in Kelvin, σ is the Stefan-Boltzmann constant, ε is the emissivity of the surface, and hrad is the radiative surface heat transfer coefficient.

Emissivity Dull black surfaces normally have emissivities of between 0.9 and 1.0. Shiny metal surfaces have very low emissivities of less than 0.1. The radiation emitted by objects at body temperature is in the infrared region of the spectrum. In this region, almost all surfaces (skin included) other than mirror-polished metallic surfaces, have an emissivity of about 0.9 to 1.0.

Energy Conservation The first law of thermodynamics is about energy conservation. Start by considering a physical system. It undergoes a process in which it receives heat and does work on something else. Let U be the internal stored energy and ΔU be the change in U in the process. The corresponding amount of heat input to the system is Q, and the system does work, W. For this situation:

ΔU = Q − W Q is positive when heat is transferred to the system. W is positive when work is done by the system on something else. For the human body, we need to include the energy input through the metabolism of food, E:

ΔU = Q − W + E Food energy (E) is often called calorific value and specifies in calories. In physics, J or kJ is used. It is measured by determining the heat produced when food is oxidised by combustion. This is then adjusted for how we actually metabolise food (for example, dietary fibre will burn in a calorimeter, but is not digested). Work Efficiency The work efficiency of the body is:

W m et a bolic en erg y i n pu t W = E − ΔU

η=

W is the mechanical work done by the body. If the body does not add to, or use, stored energy (ΔU is 0) then:

η=

W E

ΔU is negative if work is done when E is 0 (no food is taken). This represents a weight loss....