Content 10 Thermodynamics of gases Objectives Objectives 10.1 Heat capacity

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1 hermodynamics of gases ontent. Heat capacities. ork done by a gas.3 irst law of thermodynamics.4 Isothermal adiabatic changes Objectives (a) define heat capacity, specific heat capacity molar heat capacity (b) use the equations: =, = mc, = n.m = n p,m (c) derive use the equation for work done by a gas, = p d (d) State apply the first law of thermodynamics, = + (e) deduce the relationship = n,m from the first law of thermodynamics; (f) derive use the equation p,m,m = Objectives g) relates cp c to the degrees of freedom h) use the relationship = c p,m /c,m to identify the types of molecules i) describe the isothermal process of a gas j) use the equations p = = for adiabatic changes k) illustrate thermodynamics processes with p graphs l) derive use the expression for work done in the thermodynamic process 3 4. Heat capacity he study of relationships involving heat, mechanical work, other aspects of energy energy transfer for the. hermodynamic is any collection of objects that is convenient to regard as a unit, that may have the potential energy to exchange energy with its surroundings. rom the equation of specific heat capacity, = mc, c = specific heat capacity, m = mass of the object, = quantity of heat,. Heat capacity Since n = m/m, thus m = nm, M = molar mass, therefore = nmc = Mc = n, = molar specific heat (molar heat capacity) n = number of mole = temperature difference Heat capacity Molar specific heat (molar heat capacity) is defined as the amount of heat required to raise the temperature of mole gas by K or. Or = /n, he unit of molar specific heat is J K - mol - or J - mol -. Heat capacity Molar Specific Heat at onstant ressure, - is defined as the amount of heat required to raise the temperature of mole gas by K or at pressure. Or, n p Mc p -, c p = specific heat capacity at pressure, p =molar specific heat at pressure. 7 8

2 . Heat capacity Molar Specific Heat at onstant olume, is defined as the amount of heat required to raise the temperature of mole gas by K or at volume. Or n Mc. Heat capacity elationship between. onsider a graph as shown in figure. c. Heat capacity :specific heat capac ityat volume : molar specific heat at volume. In process B : Isochoric process ( volume), the work done is B 9. Heat capacity. hus the st law of thermodynamics becomes n B B B n B. Heat capacity. Heat capacity In process : Isobaric process ( pressure), the work done is thus the st law of thermodynamics becomes n Since B= then B n n n n 3 4 atio between or mole of an ideal gas, ( B ) onclusion or mole of an ideal gas f f B B thus Since = - thus herefore the ratio between is given by, f f f f is dimensionless always greater than unity. 5 6

3 . Heat capacity Heat is an energy transfer between a its surroundings that is the result of rom motion in the surroundings. Note the difference between a work process a heat process. In the former, there must be organized motion in the surroundings, but in the latter, the energy transfer is a result of rom motion in the surroundings.. Heat capacity Heat is a term that is used all the time in everyday life. In this context, it's okay to talk about the "amount of heat in a hot object," but in the classroom it is important that a teacher emphasize the precise meaning of this term. hot object may contain a lot of internal energy, but it does not contain heat. 7. Heat capacity hen a glass of water is placed on a hot plate, energy will spontaneously leave the hot plate cause the internal energy of the water to increase. Heat will always flow spontaneously from the at higher temperature to the at lower temperature, but heat can be made to flow in the opposite direction as well if work is done in the process. or example, a refrigerator relies on work done by its compressor to move heat from a cold freezer into a warm kitchen.. Heat capacity hrases like "as friction slowed the block, heat was generated in the sliding surface" are not difficult to find. In fact, as the block slows, the organized motion of the block does work on the sliding surface increases its internal energy ork done by a gas. ork done by a gas Signs onvention for Heat, ork, Sign of : = positive value Heat flow into the = negative value Heat flow out of the Sign of : = positive value ork done by the = negative value ork done on the. ork done by a gas. ork done by a gas igures onsider the infinitesimal (small, thus almost zero) work done by the gas () during the small expansion, dx in a cylinder with a movable piston as shown in figure below. Initial inal Gas dx 3 Suppose that the cylinder has a cross sectional area, the pressure exerted by the gas () at the piston face is. he work, d done by the gas is given by thus dx d d d d d Initial inal Gas dx dx cos dx 4

4 . ork done by a gas In a finite change of volume from to, d then d d : work done :gas pressure :initial volume :final volume Initial inal. ork done by a gas Gas or a change in volume at pressure, or a change in volume at volume, dx the work done is ork done at ressure - graph rea under the graph = ork done 5 7. ork done by a gas or an ideal gas n n n - graph n ln then d n thus ln ln n ln ork done in the ideal gas igures below show the pressure, against volume, graph (- Diagram) rea under the graph = ork done irst law of thermodynamics.3 irst law of thermodynamics generalisation of the principle of conservation of energy to include energy transfer through heat as well as mechanical work. Energy can be neither created nor destroyed but only transformed Heat processes work processes account for all possible energy transfers to a. It therefore follows from conservation of energy that the total change in the internal energy of a is the sum of the work done on the the heat transferred to the irst law of thermodynamics = on + into O = + If is positive if positive work is done on the. If the. he first law of thermodynamics: E int = +, E int : hange in the internal energy of the : Energy transferred by heat to the : ork done on the 3 3

5 .3 irst law of thermodynamics he first law of thermodynamics is a special case of the law of conservation of energy that relates the change in internal energy of a to the net transfer of energy by heat work. is a state variable (like,, ): its value is determined by the state of the, independent of the path..3 irst law of thermodynamics Some special cases: Isolated : one that does not interact with its surroundings. Since = =, E int = E int, i = E int, f. yclic process: a process that starts ends at the same state. Since E int =, = - while it does work,, the internal energy, changes by amount equal to = =, = initial internal energy = final internal energy = quantity of heat transferred = work done or infinitesimal (almost zero) change in energy, d = d - d 4 Ideal gas is a function only of temperature, not of, separately. (ideal gas) = kinetic energy of molecules + internal rotational vibrational energy. his is independent of the volume. lace an ideal gas in a container of volume. Suddenly double the volume (open a valve to a nd container), kinetic energy of gas does not change, the gas does no work he change in internal energy of a during any thermodynamic process is independent of path. Example: the goes from state to state as shown below. he internal energy depends only on temperature of the. If the initial final (state) temperature of the is the same, hence = = because f n he change in internal energy also zero in cyclic thermodynamic process (repeated process) because the initial final state of the is the same..4 Isothermal adiabiatic changes emperature does not change, is unchanged. 39 4

6 .4 Isothermal adiabiatic changes our specific kinds of thermodynamic processes: Isothermal process: temperature diabatic process: No heat enters/leaves a (gas) Isochoric (Isometric) process: volume Isobaric process: pressure Isothermal change Note that: If the gas exp isothermally, thus > = positive value If the gas compress isothermally, thus < = negative value Example: the compression stroke in an internal combustion engine is an approximately adiabatic process Isothermal change Isothermal rocess is defined as a process that occurs at temperature i.e. =, thus = = ; = hen n ln therefore =, then n ln Definition is defined as a process that occurs without heat transfer into or out of a i.e. thus = = = or diabatic expansion ( > ), = positive value but =negative value hence the internal energy of the decrease. or diabatic compression ( < ), = negative value but =positive value hence the internal energy of the increase. Isochoric rocess Definition is defined as a process that occurs at volume, = (no work done) thus = = = In an isochoric process, all the energy added as heat remains in the as an increase in internal energy (because the temperature of the increase). Example: heating a gas in a closed volume container Isobaric rocess Definition is defined as a process that occurs at pressure i.e. = = ( ) thus = = ( ) Example: boiling water at pressure. 45 ressure-olume Diagram (graph) he figure shows a diagram for each thermodyna mic process for a amount of an ideal gas

7 onsider the st law of thermodynamics, written in differential (infinitesimal, approaching zero) form : d = d d d = n v d hen n v d + d = d d = Because adiabatic process. hus n v d + d = rom the equation of state for an ideal gas, = n then = n/ herefore n v d + (n/)d = n d d d n d d d d d d d or finite changes in temperature volume, integrate d d d d Hence ln ln ln ln ln ork Done in diabatic rocess In adiabatic process, = hence the st law of thermodynamics becomes = - = n v = -n v = -n v ( ) = n v ( ) se equation of state, thus Summary n n Note that: diabatic expansion (d>) always occurs with a drop in temperature (d<). diabatic compression (d<) always occurs with a rise in temperature (d>). earrange the equation of state for an ideal gas : thus n n or Hence n n n but hermodynamics of Gases Heat capacities v,m = d/d p,m - v,m = = p,m / v,m irst Law of thermodynamic = + = n v,m ork done = p d = area under p- graph End of hapter Isothermal process = p = p = diabatic process = p = - = 55 56