#set document(title: "3.6 Adiabatic Processes for an Ideal Gas", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 3.6#h(0.6em)Adiabatic Processes for an Ideal Gas When an ideal gas is compressed adiabatically #math.equation(block: false, alt: "open parenthesis Q equals 0 close parenthesis ,")[$( Q = 0 ) ,$] work is done on it and its temperature increases; in an #strong[adiabatic expansion], the gas does work and its temperature drops. #strong[Adiabatic compressions] actually occur in the cylinders of a car, where the compressions of the gas-air mixture take place so quickly that there is no time for the mixture to exchange heat with its environment. Nevertheless, because work is done on the mixture during the compression, its temperature does rise significantly. In fact, the temperature increases can be so large that the mixture can explode without the addition of a spark. Such explosions, since they are not timed, make a car run poorly—it usually “knocks.” Because ignition temperature rises with the octane of gasoline, one way to overcome this problem is to use a higher-octane gasoline. Another interesting adiabatic process is the free expansion of a gas. shows a gas confined by a membrane to one side of a two-compartment, thermally insulated container. When the membrane is punctured, gas rushes into the empty side of the container, thereby expanding freely. Because the gas expands “against a vacuum” #math.equation(block: false, alt: "open parenthesis p equals 0 close parenthesis")[$( p = 0 )$], it does no work, and because the vessel is thermally insulated, the expansion is adiabatic. With #math.equation(block: false, alt: "Q equals 0")[$Q = 0$] and #math.equation(block: false, alt: "W equals 0")[$W = 0$] in the first law, #math.equation(block: false, alt: "Δ E sub int equals 0 ,")[$"Δ" E_("int") = 0 ,$] so #math.equation(block: false, alt: "E sub int sub i equals E sub int sub f")[$E_("int") i = E_("int") f$] for the free expansion. #figure(figph[The figure on the left is an illustration of the initial equilibrium state of a container with a partition in the middle dividing it into two chambers. The outer walls are insulated. The chamber on the left is full of gas, indicated by blue shading and many small dots representing the gas molecules. The right chamber is empty. The figure on the right is an illustration of the final equilibrium state of the container. The partition has a hole in it. The entire container, on both sides of the partition, is full of gas, indicated by blue shading and many small dots representing the gas molecules. The dots in the second, final equilibrium state, illustration are less dense than in the first, initial state illustration.], alt: "The figure on the left is an illustration of the initial equilibrium state of a container with a partition in the middle dividing it into two chambers. The outer walls are insulated. The chamber on the left is full of gas, indicated by blue shading and many small dots representing the gas molecules. The right chamber is empty. The figure on the right is an illustration of the final equilibrium state of the container. The partition has a hole in it. The entire container, on both sides of the partition, is full of gas, indicated by blue shading and many small dots representing the gas molecules. The dots in the second, final equilibrium state, illustration are less dense than in the first, initial state illustration.", caption: [The gas in the left chamber expands freely into the right chamber when the membrane is punctured.]) If the gas is ideal, the internal energy depends only on the temperature. Therefore, when an ideal gas expands freely, its temperature does not change. A quasi-static, adiabatic expansion of an ideal gas is represented in , which shows an insulated cylinder that contains an ideal gas. The gas is made to expand quasi-statically by removing one grain of sand at a time from the top of the piston. When the gas expands by #emph[dV], the change in its temperature is #emph[dT]. The work done by the gas in the expansion is #math.equation(block: false, alt: "d W equals p d V ; d Q equals 0")[$d W = p d V ; d Q = 0$] because the cylinder is insulated; and the change in the internal energy of the gas is, from , #math.equation(block: false, alt: "d E sub int equals n C sub V d T .")[$d E_("int") = n C_(V) d T .$] Therefore, from the first law, #math.equation(block: true, alt: "n C sub V d T equals 0 minus p d V equals − p d V ,")[$n C_(V) d T = 0 − p d V = "−" p d V ,$] so #math.equation(block: true, alt: "d T equals minus the fraction p d V over n C sub V .")[$d T = − frac(p d V, n C_(V)) .$] #figure(figph[The figure is an illustration of a container. The walls and bottom are filled with a thick layer of insulation. The chamber of the container is closed from above by a piston. Inside the chamber is a gas. There is a pile of sand on top of the piston, and a hand with tweezers is removing grains from the pile.], alt: "The figure is an illustration of a container. The walls and bottom are filled with a thick layer of insulation. The chamber of the container is closed from above by a piston. Inside the chamber is a gas. There is a pile of sand on top of the piston, and a hand with tweezers is removing grains from the pile.", caption: [When sand is removed from the piston one grain at a time, the gas expands adiabatically and quasi-statically in the insulated vessel.]) Also, for an ideal gas, #math.equation(block: true, alt: "d open parenthesis p V close parenthesis equals d open parenthesis n R T close parenthesis ,")[$d ( p V ) = d ( n R T ) ,$] so #math.equation(block: true, alt: "p d V plus V d p equals n R d T")[$p d V + V d p = n R d T$] and #math.equation(block: true, alt: "d T equals the fraction p d V plus V d p over n R .")[$d T = frac(p d V + V d p, n R) .$] We now have two equations for #emph[dT]. Upon equating them, we find that #math.equation(block: true, alt: "n C sub V V d p plus open parenthesis n C sub V plus n R close parenthesis p d V equals 0 .")[$n C_(V) V d p + ( n C_(V) + n R ) p d V = 0 .$] Now, we divide this equation by #emph[npV] and use #math.equation(block: false, alt: "C sub p equals C sub V plus R")[$C_(p) = C_(V) + R$]. We are then left with #math.equation(block: true, alt: "C sub V the fraction d p over p plus C sub p the fraction d V over V equals 0 ,")[$C_(V) frac(d p, p) + C_(p) frac(d V, V) = 0 ,$] which becomes #math.equation(block: true, alt: "the fraction d p over p plus γ the fraction d V over V equals 0 ,")[$frac(d p, p) + γ frac(d V, V) = 0 ,$] where we define #math.equation(block: false, alt: "γ")[$"γ"$] as the ratio of the molar heat capacities: #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "γ equals the fraction C sub p over C sub V .")[$γ = frac(C_(p), C_(V)) .$] ] Thus, #math.equation(block: true, alt: "∫ the fraction d p over p plus γ ∫ the fraction d V over V equals 0")[$display(∫ frac(d p, p) + γ display(∫ frac(d V, V) = 0))$] and #math.equation(block: true, alt: "ln p plus γ ln V equals constant .")[$"ln" #h(0.2em) p + "γ" "ln" #h(0.2em) V = "constant" "."$] Finally, using #math.equation(block: false, alt: "ln open parenthesis A to the power x close parenthesis equals x ln A and ln A B equals ln A plus ln B")[$"ln" ( A^(x) ) = x "ln" A #h(0.2em) "and" #h(0.2em) "ln" #h(0.2em) A B = "ln" #h(0.2em) A + "ln" #h(0.2em) B$], we can write this in the form #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "p V to the power γ equals constant .")[$p V^(γ) = "constant" "."$] ] This equation is the condition that must be obeyed by an ideal gas in a quasi-static adiabatic process. For example, if an ideal gas makes a quasi-static adiabatic transition from a state with pressure and volume #math.equation(block: false, alt: "p sub 1")[$p_(1)$] and #math.equation(block: false, alt: "V sub 1")[$V_(1)$] to a state with #math.equation(block: false, alt: "p sub 2")[$p_(2)$] and #math.equation(block: false, alt: "V sub 2 ,")[$V_(2) ,$] then it must be true that #math.equation(block: false, alt: "p sub 1 V 1 γ equals p sub 2 V 2 γ .")[$p_(1) V_(1)^(γ) = p_(2) V_(2)^(γ) .$] The adiabatic condition of can be written in terms of other pairs of thermodynamic variables by combining it with the ideal gas law. In doing this, we find that #math.equation(block: true, alt: "p to the power 1 minus γ T to the power γ equals constant")[$p^(1 − γ) T^(γ) = "constant"$] and #math.equation(block: true, alt: "T V to the power γ minus 1 equals constant .")[$T V^(γ − 1) = "constant" "."$] A reversible adiabatic expansion of an ideal gas is represented on the #emph[pV] diagram of . The slope of the curve at any point is #math.equation(block: true, alt: "the fraction d p over d V equals the fraction d over d V open parenthesis the fraction constant over V to the power γ close parenthesis equals − γ the fraction p over V .")[$frac(d p, d V) = frac(d, d V) ( frac("constant", V^(γ)) ) = "−" γ frac(p, V) .$] #figure(figph[The figure is a plot of pressure, p on the vertical axis as a function of volume, V on the horizontal axis. Two curves are plotted. Both are monotonically decreasing and concave up. One is slightly higher and has a greater curvature. This curve is labeled “isothermal.” The second curve is below the isothermal curve and has a slightly smaller curvature. This curve is labeled “adiabatic.”], alt: "The figure is a plot of pressure, p on the vertical axis as a function of volume, V on the horizontal axis. Two curves are plotted. Both are monotonically decreasing and concave up. One is slightly higher and has a greater curvature. This curve is labeled “isothermal.” The second curve is below the isothermal curve and has a slightly smaller curvature. This curve is labeled “adiabatic.”", caption: [Quasi-static adiabatic and isothermal expansions of an ideal gas.]) The dashed curve shown on this #emph[pV] diagram represents an isothermal expansion where #emph[T] (and therefore #emph[pV]) is constant. The slope of this curve is useful when we consider the second law of thermodynamics in the next chapter. This slope is #math.equation(block: true, alt: "the fraction d p over d V equals the fraction d over d V the fraction n R T over V equals minus the fraction p over V .")[$frac(d p, d V) = frac(d, d V) #h(0.2em) frac(n R T, V) = − frac(p, V) .$] Because #math.equation(block: false, alt: "γ greater than 1 ,")[$"γ" > 1 ,$] the isothermal curve is not as steep as that for the adiabatic expansion. The example Compression of an Ideal Gas in an Automobile Engine, drawn as a pV diagram: 240 cm³ of gas–air mixture at 1.00 × 10⁵ N/m² is compressed to 40 cm³. The solid curve is the adiabat pV^γ = constant, p = p₁(V₁/V)^γ; the dashed curve is the isotherm pV = constant through the same starting state. The horizontal axis is volume in tens of cm³ (x = 24 is 240 cm³) and the vertical axis is pressure in units of 1.00 × 10⁵ N/m². At the example's diatomic γ = 1.40 the compression ends at 12.3 on the adiabat but only at 6.0 on the isotherm; that gap is the pressure the mixture gains from being heated, and the same γ carries its temperature from 293 K to 293 × 6^0.40 = 600 K. Drag γ up to 1.67, the monatomic value, and the adiabat steepens; drag it down to 1 and it lies exactly on top of the isotherm. The two curves separate only because γ \> 1, which is the whole content of the sentence above.{"functions":\[{"color":"\#1f4e79","domain":\[4,24\],"expression":"(24/x)^ga","expression2":"","id":"adiabat","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#b8442a","domain":\[4,24\],"expression":"24/x","expression2":"","id":"isotherm","inequality":"lt","lineDash":\[6,4\],"lineWidth":2,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true}\],"grid":{"axisColor":"\#333333","gridColor":"\#cccccc","majorSpacing":2,"minorGridColor":"\#eeeeee","showAxes":true,"showLabels":true},"parameters":\[{"id":"up2-3-6-ga","label":"Heat-capacity ratio \\u03b3 = Cp/CV (1.40 for the example's diatomic mixture)","max":1.67,"min":1,"name":"ga","step":0.01,"unit":"","value":1.4}\],"title":"Adiabatic and isothermal compression of an ideal gas from 240 cm\\u00b3 to 40 cm\\u00b3","version":1,"viewport":{"centerX":14,"centerY":7.4,"scale":30}}#examplebox("Example 1")[Compression of an Ideal Gas in an Automobile Engine][ Gasoline vapor is injected into the cylinder of an automobile engine when the piston is in its expanded position. The temperature, pressure, and volume of the resulting gas-air mixture are #math.equation(block: false, alt: "20 ° C")[$20 #h(0.2em) "°" "C"$], #math.equation(block: false, alt: "1.00 times 10 to the power 5 N/m squared ,")[$1.00 #h(0.2em) × #h(0.2em) 10^(5) attach(#h(0.2em) "N/m", t: 2) ","$] and #math.equation(block: false, alt: "240 cm cubed")[$240 attach(#h(0.2em) "cm", t: 3)$], respectively. The mixture is then compressed adiabatically to a volume of #math.equation(block: false, alt: "40 cm cubed")[$40 attach(#h(0.2em) "cm", t: 3)$]. Note that in the actual operation of an automobile engine, the compression is not quasi-static, although we are making that assumption here. A second, important approximation is that although the vapor is a gasoline-air mixture of nitrogen, oxygen, carbon dioxide, water, and hydrocarbons, we treat it as an ideal, diatomic gas. (a) What are the pressure and temperature of the mixture after the compression? (b) How much work is done by the mixture during the compression? Strategy Because we are modeling the process as a quasi-static adiabatic compression of an ideal gas, we have #math.equation(block: false, alt: "p V to the power γ equals constant")[$p V^(γ) = #h(0.2em) "constant"$] and #math.equation(block: false, alt: "p V equals n R T")[$p V = n R T$]. The work needed can then be evaluated with #math.equation(block: false, alt: "W equals ∫ V sub 1 V sub 2 p d V")[$W = display(∫_(V_(1))^(V_(2)) p d V)$]. Solution + For an adiabatic compression we have #math.equation(block: true, alt: "p sub 2 equals p sub 1 open parenthesis the fraction V sub 1 over V sub 2 close parenthesis to the power γ .")[$p_(2) = p_(1) attach(( frac(V_(1), V_(2)) ), t: γ) .$] Since we are treating the vapor as an ideal, diatomic gas, we can use #math.equation(block: false, alt: "γ equals the fraction 7 over 5 equals 1.4")[$γ = frac(7, 5) = 1.4$]. So after the compression, the pressure of the mixture is #math.equation(block: true, alt: "p sub 2 equals open parenthesis 1.00 times 10 to the power 5 N/m squared close parenthesis open parenthesis the fraction 240 times 10 to the power −6 m cubed over 40 times 10 to the power −6 m cubed close parenthesis to the power 1.40 equals 1.23 times 10 to the power 6 N/m squared .")[$p_(2) = ( 1.00 #h(0.2em) × #h(0.2em) 10^(5) attach(#h(0.2em) "N/m", t: 2) ) attach(( frac(240 #h(0.2em) × #h(0.2em) 10^(−6) attach(#h(0.2em) "m", t: 3), 40 #h(0.2em) × #h(0.2em) 10^(−6) attach(#h(0.2em) "m", t: 3)) ), t: 1.40) = 1.23 #h(0.2em) × #h(0.2em) 10^(6) attach(#h(0.2em) "N/m", t: 2) "."$] From the ideal gas law, the temperature of the mixture after the compression is #math.equation(block: true, alt: "T sub 2, equals open parenthesis the fraction p sub 2 V sub 2 over p sub 1 V sub 1 close parenthesis T sub 1; equals the fraction open parenthesis 1.23 times 10 to the power 6 N/m squared close parenthesis open parenthesis 40 times 10 to the power −6 m cubed close parenthesis over open parenthesis 1.00 times 10 to the power 5 N/m squared close parenthesis open parenthesis 240 times 10 to the power −6 m cubed close parenthesis times 293 K; equals 600 K equals 328 ° C .")[$T_(2) & = ( frac(p_(2) V_(2), p_(1) V_(1)) ) T_(1) \ & = frac(( 1.23 #h(0.2em) × #h(0.2em) 10^(6) attach(#h(0.2em) "N/m", t: 2) ) ( 40 #h(0.2em) × #h(0.2em) 10^(−6) attach(#h(0.2em) "m", t: 3) ), ( 1.00 #h(0.2em) × #h(0.2em) 10^(5) attach(#h(0.2em) "N/m", t: 2) ) ( 240 #h(0.2em) × #h(0.2em) 10^(−6) attach(#h(0.2em) "m", t: 3) )) · 293 #h(0.2em) "K" \ & = 600 #h(0.2em) "K" = 328 #h(0.2em) "°" "C" "."$] + The work done by the mixture during the compression is #math.equation(block: true, alt: "W equals ∫ V sub 1 V sub 2 p d V .")[$W = display(∫_(V_(1))^(V_(2)) p d V) .$] With the adiabatic condition of , we may write #emph[p] as #math.equation(block: false, alt: "K / V to the power γ ,")[$K "/" V^("γ") ,$] where #math.equation(block: false, alt: "K equals p sub 1 V 1 γ equals p sub 2 V 2 γ .")[$K = p_(1) V_(1)^(γ) = p_(2) V_(2)^(γ) .$] The work is therefore #math.equation(block: true, alt: "W, equals ∫ V sub 1 V sub 2 the fraction K over V to the power γ d V; equals the fraction K over 1 minus γ open parenthesis the fraction 1 over V sub 2 to the power γ minus 1 minus the fraction 1 over V sub 1 to the power γ minus 1 close parenthesis; equals the fraction 1 over 1 minus γ open parenthesis the fraction p sub 2 V 2 γ over V sub 2 to the power γ minus 1 minus the fraction p sub 1 V 1 γ over V sub 1 to the power γ minus 1 close parenthesis; equals the fraction 1 over 1 minus γ open parenthesis p sub 2 V sub 2 minus p sub 1 V sub 1 close parenthesis; equals the fraction 1 over 1 minus 1.40 [ open parenthesis 1.23 times 10 to the power 6 N/m squared close parenthesis open parenthesis 40 times 10 to the power −6 m cubed close parenthesis; minus open parenthesis 1.00 times 10 to the")[$W & #h(0.2em) = display(∫_(V_(1))^(V_(2)) frac(K, V^(γ))) d V \ & #h(0.2em) = frac(K, 1 − γ) ( frac(1, V_(2) γ − 1) − frac(1, V_(1) γ − 1) ) \ & = #h(0.2em) frac(1, 1 − γ) ( frac(p_(2) V_(2)^(γ), V_(2) γ − 1) − frac(p_(1) V_(1)^(γ), V_(1) γ − 1) ) \ & = #h(0.2em) frac(1, 1 − γ) ( p_(2) V_(2) − p_(1) V_(1) ) \ & #h(0.2em) = frac(1, 1 − 1.40) [ ( 1.23 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "N/m"^(2) ) ( 40 #h(0.2em) × #h(0.2em) 10^(−6) #h(0.2em) "m"^(3) ) \ & − ( 1.00 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "N/m"^(2) ) ( 240 #h(0.2em) × #h(0.2em) 10^(−6) "m"^(3) ) ] \ & #h(0.2em) = −63 #h(0.2em) "J" "."$] Significance The negative sign on the work done indicates that the piston does work on the gas-air mixture. The engine would not work if the gas-air mixture did work on the piston. ] === Summary - A quasi-static adiabatic expansion of an ideal gas produces a steeper pV curve than that of the corresponding isotherm. - A realistic expansion can be adiabatic but rarely quasi-static. === Key Equations #figure(table( columns: 2, align: left, inset: 6pt, table.header([Equation of state for a closed system], [#math.equation(block: false, alt: "f open parenthesis p , V , T close parenthesis equals 0")[$f ( p , V , T ) = 0$]]), [Net work for a finite change in volume], [#math.equation(block: false, alt: "W equals ∫ V sub 1 V sub 2 p d V")[$W = display(∫_(V_(1))^(V_(2)) p d V)$]], [Internal energy of a system (average total energy)], [#math.equation(block: false, alt: "E sub int equals ∑ i open parenthesis K ¯ sub i plus U ¯ sub i close parenthesis ,")[$E_("int") = display(limits(∑)_(i) ( overline(K)_(i) + overline(U)_(i) )) ,$]], [Internal energy of a monatomic ideal gas], [#math.equation(block: false, alt: "E sub int equals n N sub A open parenthesis the fraction 3 over 2 k sub B T close parenthesis equals the fraction 3 over 2 n R T")[$E_("int") = n N_("A") ( frac(3, 2) k_("B") T ) = frac(3, 2) n R T$]], [First law of thermodynamics], [#math.equation(block: false, alt: "Δ E sub int equals Q minus W")[$"Δ" E_("int") = Q − W$]], [Molar heat capacity at constant pressure], [#math.equation(block: false, alt: "C sub p equals C sub V plus R")[$C_(p) = C_(V) + R$]], [Ratio of molar heat capacities], [#math.equation(block: false, alt: "γ equals C sub p / C sub V")[$γ = C_(p) "/" C_(V)$]], [Condition for an ideal gas in a quasi-static adiabatic process], [#math.equation(block: false, alt: "p V to the power γ equals constant")[$p V^(γ) = #h(0.2em) "constant"$]], )) === Conceptual Questions Is it possible for #math.equation(block: false, alt: "γ")[$"γ"$] to be smaller than unity? #solutionbox[ No, it is always greater than 1. ] Would you expect #math.equation(block: false, alt: "γ")[$"γ"$] to be larger for a gas or a solid? Explain. There is no change in the internal energy of an ideal gas undergoing an isothermal process since the internal energy depends only on the temperature. Is it therefore correct to say that an isothermal process is the same as an adiabatic process for an ideal gas? Explain your answer. #solutionbox[ An adiabatic process has a change in temperature but no heat flow. The isothermal process has no change in temperature but has heat flow. ] Does a gas do any work when it expands adiabatically? If so, what is the source of the energy needed to do this work? === Problems A monatomic ideal gas undergoes a quasi-static adiabatic expansion in which its volume is doubled. How is the pressure of the gas changed? #solutionbox[ pressure decreased by 0.31 times the original pressure ] An ideal gas has a pressure of 0.50 atm and a volume of 10 L. It is compressed adiabatically and quasi-statically until its pressure is 3.0 atm and its volume is 2.8 L. Is the gas monatomic, diatomic, or polyatomic? Pressure and volume measurements of a dilute gas undergoing a quasi-static adiabatic expansion are shown below. Plot ln p vs. ln V and determine #math.equation(block: false, alt: "γ")[$"γ"$] for this gas from your graph. #figure(table( columns: 2, align: left, inset: 6pt, table.header([P (atm)], [V (L)]), [20.0], [1.0], [17.0], [1.1], [14.0], [1.3], [11.0], [1.5], [8.0], [2.0], [5.0], [2.6], [2.0], [5.2], [1.0], [8.4], )) #solutionbox[ #figure(figph[The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in liters on the horizontal axis. The horizontal volume scale runs from 0 to 3.0, and the vertical pressure scale runs from 0 to 2.0. The data from the previous table is plotted as points and the equation L n (p) equals minus 1.40 L n (V) plus 3.00 is plotted as a line. The points all lie on or very close to the line.], alt: "The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in liters on the horizontal axis. The horizontal volume scale runs from 0 to 3.0, and the vertical pressure scale runs from 0 to 2.0. The data from the previous table is plotted as points and the equation L n (p) equals minus 1.40 L n (V) plus 3.00 is plotted as a line. The points all lie on or very close to the line.", caption: none) ; #linebreak() #math.equation(block: true, alt: "γ equals 0.713")[$γ = 0.713$] ] An ideal monatomic gas at 300 K expands adiabatically and reversibly to twice its volume. What is its final temperature? An ideal diatomic gas at 80 K is slowly compressed adiabatically and reversibly to half its volume. What is its final temperature? #solutionbox[ 106 K ] An ideal diatomic gas at 80 K is slowly compressed adiabatically to one-third its original volume. What is its final temperature? Compare the charge in internal energy of an ideal gas for a quasi-static adiabatic expansion with that for a quasi-static isothermal expansion. What happens to the temperature of an ideal gas in an adiabatic expansion? #solutionbox[ An adiabatic expansion has less work done and no heat flow, thereby a lower internal energy comparing to an isothermal expansion which has both heat flow and work done. Temperature decreases during adiabatic expansion. ] The temperature of n moles of an ideal gas changes from #math.equation(block: false, alt: "T sub 1")[$T_(1)$] to #math.equation(block: false, alt: "T sub 2")[$T_(2)$] in a quasi-static adiabatic transition. Show that the work done by the gas is given by #math.equation(block: true, alt: "W equals the fraction n R over γ minus 1 open parenthesis T sub 1 minus T sub 2 close parenthesis .")[$W = frac(n R, γ − 1) ( T_(1) − T_(2) ) .$] A dilute gas expands quasi-statically to three times its initial volume. Is the final gas pressure greater for an isothermal or an adiabatic expansion? Does your answer depend on whether the gas is monatomic, diatomic, or polyatomic? #solutionbox[ Isothermal has a greater final pressure and does not depend on the type of gas. ] (a) An ideal gas expands adiabatically from a volume of #math.equation(block: false, alt: "2.0 times 10 to the power −3 m cubed")[$2.0 #h(0.2em) × #h(0.2em) 10^(−3) attach(#h(0.2em) "m", t: 3)$] to #math.equation(block: false, alt: "2.5 times 10 to the power −3 m cubed")[$2.5 #h(0.2em) × #h(0.2em) 10^(−3) attach(#h(0.2em) "m", t: 3)$]. If the initial pressure and temperature were #math.equation(block: false, alt: "5.0 times 10 to the power 5 Pa")[$5.0 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "Pa"$] and 300 K, respectively, what are the final pressure and temperature of the gas? Use #math.equation(block: false, alt: "γ equals 5 / 3")[$γ = 5 "/" 3$] for the gas. (b) In an isothermal process, an ideal gas expands from a volume of #math.equation(block: false, alt: "2.0 times 10 to the power −3 m cubed")[$2.0 #h(0.2em) × #h(0.2em) 10^(−3) attach(#h(0.2em) "m", t: 3)$] to #math.equation(block: false, alt: "2.5 times 10 to the power −3 m cubed")[$2.5 #h(0.2em) × #h(0.2em) 10^(−3) attach(#h(0.2em) "m", t: 3)$]. If the initial pressure and temperature were #math.equation(block: false, alt: "5.0 times 10 to the power 5 Pa")[$5.0 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "Pa"$] and 300 K, respectively, what are the final pressure and temperature of the gas? On an adiabatic process of an ideal gas pressure, volume and temperature change such that #math.equation(block: false, alt: "p V to the power γ")[$p V^(γ)$] is constant with #math.equation(block: false, alt: "γ equals 5 / 3")[$γ = 5 "/" 3$] for monatomic gas such as helium and #math.equation(block: false, alt: "γ equals 7 / 5")[$γ = 7 "/" 5$] for diatomic gas such as hydrogen at room temperature. Use numerical values to plot two isotherms of 1 mol of helium gas using ideal gas law and two adiabatic processes mediating between them. Use #math.equation(block: false, alt: "T sub 1 equals 500 K , V sub 1 equals 1 L , and T sub 2 equals 300 K")[$T_(1) = 500 #h(0.2em) "K" , #h(0.2em) V_(1) = 1 #h(0.2em) "L" , #h(0.2em) "and" #h(0.2em) T_(2) = 300 #h(0.2em) "K"$] for your plot. #solutionbox[ #figure(figph[The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in liters on the horizontal axis. The horizontal, V, axis runs from 1.0 to 2.0. The vertical, p, axis runs from 0 to about 40. Two isotherms are shown. One isotherm is for T equal to 500 K, with the pressure starting at about 40 atmospheres when the volume is 1.0 Liter and decreasing with volume to about 25 atmospheres at 2.0 liters. The second isotherm is for T equal to 300 K, with the pressure starting at about 25 atmospheres when the volume is 1.0 Liter and decreasing with volume to a little over 10 atmospheres at 2.0 liters. A third plot, labeled “Adiabatic” starts with the 500 K isotherm, at 1.0 L and about 40 atmospheres, and ends with the 300 K isotherm, at 2.0 L and just over 10 atmospheres.], alt: "The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in liters on the horizontal axis. The horizontal, V, axis runs from 1.0 to 2.0. The vertical, p, axis runs from 0 to about 40. Two isotherms are shown. One isotherm is for T equal to 500 K, with the pressure starting at about 40 atmospheres when the volume is 1.0 Liter and decreasing with volume to about 25 atmospheres at 2.0 liters. The second isotherm is for T equal to 300 K, with the pressure starting at about 25 atmospheres when the volume is 1.0 Liter and decreasing with volume to a little over 10 atmospheres at 2.0 liters. A third plot, labeled “Adiabatic” starts with the 500 K isotherm, at 1.0 L and about 40 atmospheres, and ends with the 300 K isotherm, at 2.0 L and just over 10 atmospheres.", caption: none) ] Two moles of a monatomic ideal gas such as helium is compressed adiabatically and reversibly from a state (3 atm, 5 L) to a state with pressure 4 atm. (a) Find the volume and temperature of the final state. (b) Find the temperature of the initial state of the gas. (c) Find the work done by the gas in the process. (d) Find the change in internal energy of the gas in the process. === Additional Problems Consider the process shown below. During steps #emph[AB] and #emph[BC], 3600 J and 2400 J of heat, respectively, are added to the system. (a) Find the work done in each of the processes #emph[AB], #emph[BC], #emph[AD], and #emph[DC]. (b) Find the internal energy change in processes #emph[AB] and #emph[BC]. (c) Find the internal energy difference between states #emph[C] and #emph[A]. (d) Find the total heat added in the #emph[ADC] process. (e) From the information given, can you find the heat added in process #emph[AD]? Why or why not? #figure(figph[The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in liters on the horizontal axis. The horizontal volume scale runs from 0 to 7, and the vertical pressure scale runs from 0 to 5. Four points, A, B, C, and D, are labeled on the plot and their pressures and volumes are labeled on the axes. Point A is at volume 3 L, pressure 2 atmospheres. Point B is at volume 3 L, pressure 5 atmospheres. Point C is at volume 7 L, pressure 5 atmospheres. Point D is at volume 7 L, pressure 2 atmospheres. A path goes from A up to B and across to C. Another path goes from A across to D and then up to C.], alt: "The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in liters on the horizontal axis. The horizontal volume scale runs from 0 to 7, and the vertical pressure scale runs from 0 to 5. Four points, A, B, C, and D, are labeled on the plot and their pressures and volumes are labeled on the axes. Point A is at volume 3 L, pressure 2 atmospheres. Point B is at volume 3 L, pressure 5 atmospheres. Point C is at volume 7 L, pressure 5 atmospheres. Point D is at volume 7 L, pressure 2 atmospheres. A path goes from A up to B and across to C. Another path goes from A across to D and then up to C.", caption: none) #solutionbox[ a. #math.equation(block: false, alt: "W sub A B equals 0 , W sub B C equals 2026 J , W sub A D equals 810.4 J , W sub D C equals 0 ;")[$W_(A B) = 0 , W_(B C) = 2026 #h(0.2em) "J" , W_(A D) = 810.4 #h(0.2em) "J" , W_(D C) = 0 ;$] b. #math.equation(block: false, alt: "Δ E sub A B equals 3600 J , Δ E sub B C equals 374 J;")[$"Δ" E_(A B) = 3600 #h(0.2em) "J" , "Δ" E_(B C) = 374 #h(0.2em) "J;"$] c. #math.equation(block: false, alt: "Δ E sub A C equals 3974 J ;")[$"Δ" E_(A C) = 3974 #h(0.2em) "J" ;$] d. #math.equation(block: false, alt: "Q sub A D C equals 4784 J;")[$Q_(A D C) = 4784 #h(0.2em) "J;"$] e. No, because heat was added for both parts #emph[AD] and #emph[DC]. There is not enough information to figure out how much is from each segment of the path. ] A car tire contains #math.equation(block: false, alt: "0.0380 m cubed")[$0.0380 attach(#h(0.2em) "m", t: 3)$] of air at a pressure of #math.equation(block: false, alt: "2.20 times 10 to the power 5 Pa")[$2.20 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "Pa"$] (about 32 psi). How much more internal energy does this gas have than the same volume has at zero gauge pressure (which is equivalent to normal atmospheric pressure)? A helium-filled rigid, sealed container has a gauge pressure of 0.200 atm and a volume of 10.0 L. How much greater is the internal energy of the helium in the container than it would be at zero gauge pressure? Assume the volume doesn't change and no gas leaks in or out. #solutionbox[ 300 J ] Steam to drive an old-fashioned steam locomotive is supplied at a constant gauge pressure of #math.equation(block: false, alt: "1.75 times 10 to the power 6 N/m squared")[$1.75 #h(0.2em) × #h(0.2em) 10^(6) "N/m"^(2)$] (about 250 psi) to a piston with a 0.200-m radius. (a) By calculating #math.equation(block: false, alt: "p Δ V")[$p "Δ" V$], find the work done by the steam when the piston moves 0.800 m. Note that this is the net work output, since gauge pressure is used. (b) Now find the amount of work by calculating the force exerted times the distance traveled. Is the answer the same as in part (a)? A hand-driven tire pump has a piston with a 2.50-cm diameter and a maximum stroke of 30.0 cm. (a) How much work do you do in one stroke if the average gauge pressure is #math.equation(block: false, alt: "2.4 times 10 to the power 5 N/m squared")[$2.4 #h(0.2em) × #h(0.2em) 10^(5) attach(#h(0.2em) "N/m", t: 2)$] (about 35 psi)? (b) What average force do you exert on the piston, neglecting friction and gravitational force? #solutionbox[ a. #emph[W] = #emph[pV] = J; b. #emph[F] = #emph[pA] = 117.8 N ] Calculate the net work output of a heat engine following path #emph[ABCDA] as shown below. #figure(figph[The figure is a plot of pressure, p, in nuits of 10 to the 6 Newtons per meter squared on the vertical axis as a function of volume, V, in 10 to the minus 3 cubic meters on the horizontal axis. The horizontal volume scale runs from 0 to 4, and the vertical pressure scale runs from 0 to about 4. Four points, A, B, C, and D, are labeled on the plot and their pressures and volumes are labeled on the axes. Point A is at volume 1.0 times 10 to the -3 cubic meters, pressure 2.6 times 10 to the 6 Newtons per meter squared. Point B is at volume 4.0 times 10 to the -3 cubic meters, pressure 2.0 times 10 to the 6 Newtons per meter squared. Point C is at volume 4.0 times 10 to the -3 cubic meters, pressure 0.6 times 10 to the 6 Newtons per meter squared. Point D is at volume 1.0 times 10 to the -3 cubic meters, pressure 1.0 times 10 to the 6 Newtons per meter squared. A straight line connects A to B, another straight line B to C, another straight line C to D, and another straight line back to A.], alt: "The figure is a plot of pressure, p, in nuits of 10 to the 6 Newtons per meter squared on the vertical axis as a function of volume, V, in 10 to the minus 3 cubic meters on the horizontal axis. The horizontal volume scale runs from 0 to 4, and the vertical pressure scale runs from 0 to about 4. Four points, A, B, C, and D, are labeled on the plot and their pressures and volumes are labeled on the axes. Point A is at volume 1.0 times 10 to the -3 cubic meters, pressure 2.6 times 10 to the 6 Newtons per meter squared. Point B is at volume 4.0 times 10 to the -3 cubic meters, pressure 2.0 times 10 to the 6 Newtons per meter squared. Point C is at volume 4.0 times 10 to the -3 cubic meters, pressure 0.6 times 10 to the 6 Newtons per meter squared. Point D is at volume 1.0 times 10 to the -3 cubic meters, pressure 1.0 times 10 to the 6 Newtons per meter squared. A straight line connects A to B, another straight line B to C, another straight line C to D, and another straight line back to A.", caption: none) What is the net work output of a heat engine that follows path #emph[ABDA] in the preceding problem with a straight line from #emph[B] to #emph[D]? Why is the work output less than for path #emph[ABCDA]? #solutionbox[ #math.equation(block: true, alt: "2.4 times 10 cubed J")[$2.4 #h(0.2em) × #h(0.2em) 10^(3) #h(0.2em) "J"$] ] Five moles of a monatomic ideal gas in a cylinder at #math.equation(block: false, alt: "27 ° C")[$27 #h(0.2em) "°" "C"$] is expanded isothermally from a volume of 5 L to 10 L. (a) What is the change in internal energy? (b) How much work was done on the gas in the process? (c) How much heat was transferred to the gas? Four moles of a monatomic ideal gas in a cylinder at #math.equation(block: false, alt: "27 ° C")[$27 #h(0.2em) "°" "C"$] is expanded at constant pressure equal to 1 atm until its volume doubles. (a) What is the change in internal energy? (b) How much work was done by the gas in the process? (c) How much heat was transferred to the gas? #solutionbox[ a. 15,000 J; b. 10,000 J; c. 25,000 J ] Helium gas is cooled from #math.equation(block: false, alt: "20 ° C")[$20 #h(0.2em) "°" "C"$] to #math.equation(block: false, alt: "10 ° C")[$10 #h(0.2em) "°" "C"$] by expanding from 40 atm to 1 atm. If there is 1.4 mol of helium, (a) What is the final volume of helium? (b) What is the change in internal energy? In an adiabatic process, oxygen gas in a container is compressed along a path that can be described by the following pressure in atm as a function of volume V, with #math.equation(block: false, alt: "V sub 0 equals 1 L")[$V_(0) = 1 L$]: #math.equation(block: false, alt: "p equals open parenthesis 3.0 atm close parenthesis open parenthesis V / V sub 0 close parenthesis to the power −1.2")[$p = ( 3.0 #h(0.2em) "atm" ) ( V "/" V_(0) )^(−1.2)$]. The initial and final volumes during the process were 2 L and 1.5 L, respectively. Find the amount of work done on the gas. #solutionbox[ 78 J ] A cylinder containing three moles of a monatomic ideal gas is heated at a constant pressure of 2 atm. The temperature of the gas changes from 300 K to 350 K as a result of the expansion. Find work done (a) on the gas; and (b) by the gas. A cylinder containing three moles of nitrogen gas is heated at a constant pressure of 2 atm. The temperature of the gas changes from 300 K to 350 K as a result of the expansion. Find work done (a) on the gas, and (b) by the gas by using van der Waals equation of state instead of ideal gas law. #solutionbox[ A cylinder containing three moles of nitrogen gas is heated at a constant pressure of 2 atm. a. 1250 J; b. 1250 J ] Two moles of a monatomic ideal gas such as helium is compressed adiabatically and reversibly from a state (3 atm, 5 L) to a state with a pressure of 4 atm. (a) Find the volume and temperature of the final state. (b) Find the temperature of the initial state. (c) Find work done by the gas in the process. (d) Find the change in internal energy in the process. Assume #math.equation(block: false, alt: "C sub V equals 5 R")[$C_(V) = 5 R$] and #math.equation(block: false, alt: "C sub p equals C sub V plus R")[$C_(p) = C_(V) + R$] for the diatomic ideal gas in the conditions given. An insulated vessel contains 1.5 moles of argon at 2 atm. The gas initially occupies a volume of 5 L. As a result of the adiabatic expansion the pressure of the gas is reduced to 1 atm. (a) Find the volume and temperature of the final state. (b) Find the temperature of the gas in the initial state. (c) Find the work done by the gas in the process. (d) Find the change in the internal energy of the gas in the process. #solutionbox[ a. 7.6 L, 61.6 K; b. 81.3 K; c. #math.equation(block: false, alt: "3.63 L times atm equals 367 J")[$3.63 #h(0.2em) "L" · "atm" = 367 #h(0.2em) "J"$]; d. −367 J ] === Challenge Problems One mole of an ideal monatomic gas occupies a volume of #math.equation(block: false, alt: "1.0 times 10 to the power −2 m cubed")[$1.0 #h(0.2em) × #h(0.2em) 10^(−2) attach(#h(0.2em) "m", t: 3)$] at a pressure of #math.equation(block: false, alt: "2.0 times 10 to the power 5 N/m squared .")[$2.0 #h(0.2em) × #h(0.2em) 10^(5) attach(#h(0.2em) "N/m", t: 2) .$] (a) What is the temperature of the gas? (b) The gas undergoes a quasi-static adiabatic compression until its volume is decreased to #math.equation(block: false, alt: "5.0 times 10 to the power −3 m cubed .")[$5.0 #h(0.2em) × #h(0.2em) 10^(−3) attach(#h(0.2em) "m", t: 3) .$] What is the new gas temperature? (c) How much work is done on the gas during the compression? (d) What is the change in the internal energy of the gas? One mole of an ideal gas is initially in a chamber of volume #math.equation(block: false, alt: "1.0 times 10 to the power −2 m cubed")[$1.0 #h(0.2em) × #h(0.2em) 10^(−2) attach(#h(0.2em) "m", t: 3)$] and at a temperature of #math.equation(block: false, alt: "27 ° C")[$27 #h(0.2em) "°" "C"$]. (a) How much heat is absorbed by the gas when it slowly expands isothermally to twice its initial volume? (b) Suppose the gas is slowly transformed to the same final state by first decreasing the pressure at constant volume and then expanding it isobarically. What is the heat transferred for this case? (c) Calculate the heat transferred when the gas is transformed quasi-statically to the same final state by expanding it isobarically, then decreasing its pressure at constant volume. #solutionbox[ a. 1700 J; b. 1200 J; c. 2400 J ] A bullet of mass 10 g is traveling horizontally at 200 m/s when it strikes and embeds in a pendulum bob of mass 2.0 kg. (a) How much mechanical energy is dissipated in the collision? (b) Assuming that #math.equation(block: false, alt: "C sub v")[$C_(v)$] for the bob plus bullet is 3R, calculate the temperature increase of the system due to the collision. Take the molecular mass of the system to be 200 g/mol. The insulated cylinder shown below is closed at both ends and contains an insulating piston that is free to move on frictionless bearings. The piston divides the chamber into two compartments containing gases A and B. Originally, each compartment has a volume of #math.equation(block: false, alt: "5.0 times 10 to the power −2 m cubed")[$5.0 #h(0.2em) × #h(0.2em) 10^(−2) attach(#h(0.2em) "m", t: 3)$] and contains a monatomic ideal gas at a temperature of #math.equation(block: false, alt: "0 ° C")[$0 #h(0.2em) "°" "C"$] and a pressure of 1.0 atm. (a) How many moles of gas are in each compartment? (b) Heat Q is slowly added to A so that it expands and B is compressed until the pressure of both gases is 3.0 atm. Use the fact that the compression of B is adiabatic to determine the final volume of both gases. (c) What are their final temperatures? (d) What is the value of Q? #figure(figph[The figure is an illustration of a container with a partition in the middle dividing it into two chambers. A double headed horizontal arrow above the partition indicates that it is moveable The outer walls are insulated. The chamber on the left is labeled with an A, and is full of one gas, indicated by blue shading and many small dots representing the gas molecules. The right chamber is labeled with a B, and is full of a second gas, indicated by red shading and many small dots representing the gas molecules.], alt: "The figure is an illustration of a container with a partition in the middle dividing it into two chambers. A double headed horizontal arrow above the partition indicates that it is moveable The outer walls are insulated. The chamber on the left is labeled with an A, and is full of one gas, indicated by blue shading and many small dots representing the gas molecules. The right chamber is labeled with a B, and is full of a second gas, indicated by red shading and many small dots representing the gas molecules.", caption: none) #solutionbox[ a. 2.2 mol; b. #math.equation(block: false, alt: "V sub A equals 2.6 times 10 to the power −2 m cubed")[$V_(A) = 2.6 #h(0.2em) × #h(0.2em) 10^(−2) "m"^(3)$], #math.equation(block: false, alt: "V sub B equals 7.4 times 10 to the power −2 m cubed")[$V_(B) = 7.4 #h(0.2em) × #h(0.2em) 10^(−2) "m"^(3)$]; c. #math.equation(block: false, alt: "T sub A equals 1220 K , T sub B equals 430 K")[$T_(A) = 1220 #h(0.2em) "K" , T_(B) = 430 #h(0.2em) "K"$]; d. 30,500 J ] In a diesel engine, the fuel is ignited without a spark plug. Instead, air in a cylinder is compressed adiabatically to a temperature above the ignition temperature of the fuel; at the point of maximum compression, the fuel is injected into the cylinder. Suppose that air at #math.equation(block: false, alt: "20 ° C")[$20 #h(0.2em) "°" "C"$] is taken into the cylinder at a volume #math.equation(block: false, alt: "V sub 1")[$V_(1)$] and then compressed adiabatically and quasi-statically to a temperature of #math.equation(block: false, alt: "600 ° C")[$600 #h(0.2em) "°" "C"$] and a volume #math.equation(block: false, alt: "V sub 2 .")[$V_(2) .$] If #math.equation(block: false, alt: "γ equals 1.4 ,")[$γ = 1.4 ,$] what is the ratio #math.equation(block: false, alt: "V sub 1 / V sub 2 ?")[$V_(1) "/" V_(2) ?$] (Note: In an operating diesel engine, the compression is not quasi-static.)