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Suppose that an unsaturated air mass is rising and cooling

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder ISBN: 9780201380279 40

Solution for problem 44P Chapter 5

An Introduction to Thermal Physics | 1st Edition

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An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

An Introduction to Thermal Physics | 1st Edition

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Problem 44P

Problem 44P

Suppose that an unsaturated air mass is rising and cooling at the dry adiabatic lapse rate found in Problem 1. If the temperature at ground level is 25° C and the relative humidity there is 50%, at what altitude will this air mass become saturated so that condensation begins and a cloud forms (see below Figure 1)? (Refer to the vapour pressure graph drawn shown below Problem 5.)

Figure 1: Cumulus clouds form when rising air expands adiabaticallyand cools to the dew point the onset of condensation slowsthe cooling, increasing the tendency of the air to rise further.These clouds began to form in late morning, in a sky that was clear onlyan hour before the photo was taken. By mid-afternoon they had developedinto thunderstorms.

Problem 1:

In Problem 2 you calculated the pressure of earth’s atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottommost 10–15 km of the atmosphere (called the troposphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient |dT/dz| exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling.

(a) Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation

(b) Assume that dT/dz is just at the critical value for convection to begin, so that the vertical forces on a convecting air mass are always approximately in balance. Use the result of Problem 2 (b) to find a formula for dT/dz in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately - 10°C/km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.

Problem 2:

The exponential atmosphere.

(a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest , the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.

(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem 3.) Show, then, that the pressure obeys the differential equation

called the barometric equation.

(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(O)e−mgz/kT. Show also that the density obeys a similar equation.

(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft , 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/ Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)

Problem 3:

Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), 02 (21%), and argon (1%).

Problem 4:

Ordinarily, the partial pressure of water vapour in the air is less than the equilibrium vapour pressure at the ambient temperature; this is why a cup of water will spontaneously evaporate. The ratio of the partial pressure of water vapour to the equilibrium vapour pressure is called the relative humidity. When the relative humidity is 100%, so that water vapour in the atmosphere would be in diffusive equilibrium with a cup of liquid water, we say that the air is saturated. The dew point is the temperature at which the relative humidity would be 100%, for a given partial pressure of water vapour.

(a) Use the vapor pressure equation (below Problem 5) and the data in below Figure 2 to plot a graph of the vapor pressure of water from 0°C to 40° C. Notice that the vapour pressure approximately doubles for every 10° increase in temperature.

(b) The temperature on a certain summer day is 30°C. What is the dew point if the relative humidity is 90%? What if the relative humidity is 40%?

Problem 5:

The Clausius-Clapeyron below relation is a differential equation that can, in principle, be solved to find the shape of the entire phase-boundary curve. To solve it, however, you have to know how both L and ΔU depend on temperature and pressure. Often, over a reasonably small section of the curve, you can take L to be constant. Moreover, if one of the phases is a gas, you can usually neglect the volume of the condensed phase and just take ΔU to be the volume of the gas, expressed in terms of temperature and pressure using the ideal gas law. Making all these assumptions, solve the differential equation explicitly to obtain the following formula for the phase boundary curve:

P = (constant) × e–L/RT,

This result is called the vapour pressure equation. Caution: Be sure to use this formula only when all the assumptions just listed are valid.

Relation:

Figure 2: Phase diagram for H2O (not to scale). The table gives the vapor pressure and molar latent heat for the solid-gas transformation (first three entries) and the liquid-gas transformation (remaining entries). Data from Keenan et al (1978) and Lide (1994).

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Tuesday 4/5/16 Why did the United States enter the war 1. Financial commitment to the Allies a. War was economic boom for US b. Our money was going towards Britain and France 2. Shared principles of Democracy a. Opposition to German militarism b. Not everyone was pro­British 3. German attacks on American neutrality a. Sinking of the Lusitania because Germans didn’t want America and Britain to trade () b. “Unrestricted submarine warfare” beings early 1917, America declares war with Germany in April 1917 America at War: What was its contribution to victory ● America didn’t determine the strategy ● The Donkeys ( ​1961) ● German 210mm GUNS­ artillery was the big killer of soldiers in the war of 1914­1918 ● War of attrition­ US didn’t have any tactics ● US needed, “men, men and more men” ● Germans end the war agreeing to an armistice (Armistice Day, 11/11) Problems of Peacemaking Wilson at Versailles Why did the US enter the war 1. Financial commitment to the allies 2. Shared principles of democracy 3. German attacks on American neutrality And... 4. How could we influence the post­war peace, if we were not actively involved in fighting the war ● “Peace without victory,” ­­Wilson, January 1917 (before we enter the war) ● “Fourteen points” speech, ­­Wilson, January 1918 ○ End to secret treaties, establish League of Nations Wilson’s New World Order ● Spread democracy because democracies did not engage in wars of conquest ● An end to trade barriers would reduce tensions that led to war ● A “league of nations” rather than arms and alliances would be the key to international order There were two problems: One was the Europeans The reality of Versailles ● Britain would not accept freedom of the seas ● “Open diplomacy” was conducted behind closed doors ● Peace without victory became the “war guilt clause” for Germany ○ Made the Germans resentful and determined to “get even” when the opportunity presented itself in the future ● Self­determination for some; but other border realignments just created new problems The other problem was the US Opposition to the Treaty *** in textbook ● November 1918 elections gave control of Congress to the Republicans ○ President Wilson did not involve Republicans in the peace negotiations, even though he needed a Republican­dominated Senate to approve any treaty (Art II, Sec 2 of the Constitution) ● Irreconcilables­ did not want anything to do with a League of Nations ● Reservationists­ not necessarily against a League of Nations but wanted restrictions on its authority over the United States ● Opposition to Article 10­ did it commit nations to using force to maintain the peace and guarantee territorial integrity The Twenties Events of 1918­1919 ● Influenza­ killed 500,000 Americans (more American soldiers died of this than at the hands of the Germans) ● “Red Scare” generated by Bolshevik (Communist) Revolution in Russia and bombing campaign in US *** in textbook ● 1919­ more strikes than in any other year of American history ● Chicago White Sox threw the World Series! ● Manufacturing inexpensive consumer goods (electric mixer, the vacuum, refrigerator, washing machine) ● Age of consumption ● Instant gratification, fulfillment with consumption A consumer culture ● A society in which the majority of people seek fulfillment and defines identity through acts of consumption ● New values like, “instant gratification” rather than self­denial, restraint, saving for the future, and so on (the supposed values of the older generation) ● “Problems” resolved through consumption The automobile: Backbone of Industry ● 1900 → 300 firms produced 4,000 cars ● 1922 → Ford produced 2 million cars ● 1927 → one car for every 5.3 people in the US; in France one for every 44 people; in Germany one for every 196 Model T cost went down Automobiles encouraged consumption I.e. General Motors Cadillac → different styles of cars came out every year so your car could complement your personality ­ Promote dissatisfaction so people buy new cars ● Clyde Barrow (Bonnie and Clyde)­ “I always prefer to steal a Ford.” Car industry set up credit for consumers (so people can get a loan) ● Mass entertainment! ● 1929 40% houses had radios ● Film industry! Increasingly homogenized America… people dressing the same way, buying the same products, watching the same movies, but this creates tension in society because of this new culture Culture Clash *** In textbook ● The Triumph of Nativism (Immigration Restriction) ● “National Origins” or Immigration Act of 1924 ● Instituted a permanent quota system, with total immigration capped at 164,000 based on percentages (2 percent) of ethnic origins shown in 1890 census ● Example: Italy’s quota was 3,845, great Britain’s was 65,361 Thursday 4/7/16 Culture Clash ● The Triumph of nativism (Immigration Restriction) ● The Second Ku Klux Klan ○ Earlier KKK was in South, this one is in the Midwest (OH, IN, TX, OK, OR) ○ 100% Americanism­­ no more immigrants! ● The Scopes Trial ○ Can’t teach evolution­ too secular ○ Similar to Plessy vs. Ferguson­ separate but equal Politics of the 1920s Resurgent Republicanism Warren Harding (1920) ● “I can handle my enemies; it’s my damn friends I have to worry about!” ● ^^^ Corruption during Harding Administration ● Trickle­down theory of economics Calvin Coolidge (1924) ● Coolidge Joke: “Did you hear that former president Coolidge was found dead” ­­“Really! How could they tell” Herbert Hoover (1928) ● Hoover is “certainly a wonder and I wish we could make him president of the United States. There would not be a better one.” ­­ FDR The Great Depression (under the Hoover Administration) ● “Great Engineer” ● Said he would donate presidential salary to charity Why Depression ***in textbook ● The Stock Market Crash, 1929 ○ Shares decreased by about 40% ● Depressed Agricultural prices and farm closures ● Lack of diversity in economy ● Overproduction of consumer goods “... all of the policies of the New Deal failed to end the Great Depression; it ended when the United States began rearming in 1941...” ​n Economic History of the US ­ So… in order to end a depression, go to war ● Depressions happen about every 25­30 years ● But this Great Depression is the only one that doesn’t go away immediately and becomes a worldwide depression Some figures: ● National income: ○ $87.4 billion in 1929 ○ $41.7 billion in 1932 ● By 1932, 20­25% national unemployment with higher statewide numbers: ○ 50% in Cleveland ○ 80% in Toledo ● Bank closures to 1933 wiped out $7 billion in savings ● Hoover believed the Great Depression was only temporary ○ Government never did anything to help the depression and it would go away ○ Hoover believes he should do the same­ as it got worse, he looked bad ○ Democrats were trying to embarrass Hoover ○ Hoovervilles, Hoover flags, “Hard times are still Hoover­ing over us” Election of 1928­ Almost all the states Election of 1932­ only took 6 states FDR ● Had the draft of a speech against high tariffs, another supporting it, and told his speech writers to mesh the two together ● New Deal ● Willingness to try new things ● Confident grin ● Attempted assassination of FDR, but wounding others ● Fireside addresses Road to Recovery ● Bank holiday ○ Emergency Banking Act, 9 March 1933­­ this bill was passed unanimously by the house (seven “no” votes in the Senate​nread by any member ● The hundred days­­ 15 major pieces of legislation National Recovery Administration ● Government­Sanctioned cartels ● Industrial compacts and codes to set wages, hours, and working conditions ● Part of the National Industrial Recovery Act (NIRA) Civilian Conservation Corps ● A work­relief program ● 3 million young men employed; paid $30 a month (had to send $25 home) ● National forests; flood control; beautification projects Public Works Administration ● First of the major “make­work” programs of the New Deal ● Allotted $3.3 billion for public works (idea is to put money into people’s pockets quickly) JMU was built using PWA project money! Problems ● Conservatives thought that New Deal programs were corrupting “American Ideals”­­ FDR was going too far! ● Radicals saw the Great Depression as proof that capitalism was dead­­ FDR was not doing enough to recognize that reality! ● And economic indicators did not indicate that much recovery was taking place Floyd Olson of Minnesota and the Farm­Labor Party Upton Sinclair ● Democratic candidate for Governor of California in 1934 ● Epic­­ end poverty in California program; seize idle lands and factories and turn them over to workers’ and farmers’ cooperatives Huey Long of Louisiana ● “Share our wealth society” (1934) ● FDR: Long was “one of the two most dangerous men in the country.” ● DICTATORSHIP The “Second” New Deal ● Social Security Act (1935) ● WPA­­ Work Progress Administration ○ $11 billion works program (included the ex­slave interviews) ○ Nation’s single largest employer ● Wagner Act, or National Labor Relations Act (1935)

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Chapter 5, Problem 44P is Solved
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Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

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Suppose that an unsaturated air mass is rising and cooling