PHYS 402 University of Nebraska High School SEC1 Kinetic Theory & Ideal Gases HW

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PHYS 402

University of Nebraska high school

PHYS

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Temperature, Ideal Gases, and Kinetic Theory

  1. a.) The surface temperature of the sun is about 5750 K. What is this temperature in the Fahrenheit scale?
  2. (b.) One of the hottest temperatures ever recorded on the surface of Earth was 134oF in Death Valley, CA. What is this temperature in the Celsius scale? What is this temperature in the Kelvin scale?
  3. (c.) At what temperature do the Fahrenheit and Kelvin scales have the same numerical value?
    Hint: Write down one of the two equations for converting from one of these scales to the other and think about how you can make the left hand side of the equation equal to the right hand side. Try out some numerical values and/or use algebraic manipulations.

Problem 2. 7.5 mol of helium are in a 15 L cylinder. The pressure gauge on the cylinder reads 65 psi. Note: Gauge Pressure is defined as absolute pressure minus atmospheric pressure (1 atm). This is because gauge pressure is zero-referenced against ambient air pressure.

(a.) What is the temperature of the gas in oC?
(b.) What is the average kinetic energy of a helium atom?

Problem 3. Total lung capacity of a typical adult is approximately 5.0 L. Approximately 20% of the air is oxygen. At sea level and at an average body temperature of 37 oC, how many moles of oxygen do the lungs contain at the end of an inhalation?

Problem 4. The gauge pressure in your car tires is 2.50 × 105 N/m2 at a temperature of 35.0 oC when you drive onto a ferry boat to Alaska. What is their gauge pressure later, when their temperature has dropped to −40.0 oC? Recall: Gauge Pressure is defined as absolute pressure minus atmospheric pressure (1 atm).

Problem 5. Much of the gas near the Sun is atomic hydrogen. (Note that the atomic mass of hydrogen is m(1H)=1u=1.66×10−27 kg)

(a.) What is the average kinetic energy (in Joules) of hydrogen atoms on the 5500oC surface of the Sun? (b.) What is the average velocity vrms of the hydrogen atoms?

Problem 6. An ideal gas is at 20 oC.

(a.) (b.)

2

If we double the average kinetic energy of the ga

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Homework Set 1 PHYS 402 Sec 01 Summer 2020 Due Date: Monday, June 29, 2020; 8:10 AM This homework assignment is to be completed and submitted individually, but collaboration is allowed. Show all of your work - and be detailed about it! My primary concern is that you can show that you understand the material, not that you can write down the correct numerical answer. The only way I can see what you are thinking is for you to write it down. So, the more you show your work and your thoughts on paper, the better you will most likely do on this assignment. 1 Temperature, Ideal Gases, and Kinetic Theory Universal Gas Constant: R = 8.31 J/(mol · K) = 1.99 cal/(mol · K) = 0.0821 L/(mol · K) Problem 1. (a.) The surface temperature of the sun is about 5750 K. What is this temperature in the Fahrenheit scale? (b.) One of the hottest temperatures ever recorded on the surface of Earth was 134o F in Death Valley, CA. What is this temperature in the Celsius scale? What is this temperature in the Kelvin scale? (c.) At what temperature do the Fahrenheit and Kelvin scales have the same numerical value? Hint: Write down one of the two equations for converting from one of these scales to the other and think about how you can make the left hand side of the equation equal to the right hand side. Try out some numerical values and/or use algebraic manipulations. Problem 2. 7.5 mol of helium are in a 15 L cylinder. The pressure gauge on the cylinder reads 65 psi. Note: Gauge Pressure is defined as absolute pressure minus atmospheric pressure (1 atm). This is because gauge pressure is zero-referenced against ambient air pressure. (a.) What is the temperature of the gas in o C? (b.) What is the average kinetic energy of a helium atom? Problem 3. Total lung capacity of a typical adult is approximately 5.0 L. Approximately 20% of the air is oxygen. At sea level and at an average body temperature of 37 o C, how many moles of oxygen do the lungs contain at the end of an inhalation? Problem 4. The gauge pressure in your car tires is 2.50 × 105 N/m2 at a temperature of 35.0 o C when you drive onto a ferry boat to Alaska. What is their gauge pressure later, when their temperature has dropped to −40.0 o C? Recall: Gauge Pressure is defined as absolute pressure minus atmospheric pressure (1 atm). Problem 5. Much of the gas near the Sun is atomic hydrogen. (Note that the atomic mass of hydrogen is m(1 H) = 1 u = 1.66 × 10−27 kg) (a.) What is the average kinetic energy (in Joules) of hydrogen atoms on the 5500o C surface of the Sun? (b.) What is the average velocity vrms of the hydrogen atoms? Problem 6. An ideal gas is at 20 o C. (a.) If we double the average kinetic energy of the gas atoms, what is the new temperature in o C? (b.) What if, instead, the gas is cooled, reducing the average kinetic energy by 10%. What is the new temperature in o C? 2 Heat and Heat Transfer Unless otherwise specified, please refer to the tables in Chapter 14 of your textbook for specific heats, heats of fusion and vaporization, and thermal conductivities of various substances. Problem 7. (a.) How much energy must be removed from a 200 g block of ice to cool it from 0 o C to −30 o C? (b.) A 0.250 kg block of a pure material is heated from 20.0 o C to 65.0 o C by the addition of 4.35 kJ of energy. Calculate its specific heat and identify (using the table in your textbook) the substance of which it is most likely composed. Problem 8. (a.) How much heat is needed to change 20 g of mercury at 20 o C into mercury vapor at the boiling point (357o C)? (b.) 100 J of heat energy are transferred to 20 g of mercury initially at 20 o C. By how much does the temperature increase? (c.) How much heat is needed to raise the temperature of 20 g of water by the same amount? Problem 9. It is important for the body to have mechanisms to effectively cool itself; if not, moderate exercise could easily increase body temperatures to dangerous levels. Suppose a 70 kg man runs on a treadmill for 30 min, using a metabolic power of 1000 W . Assume that all of this power goes to thermal energy in the body. If he couldn’t perspire or otherwise cool his body, by how much would his body temperature rise during this exercise? Recall: Power is defined as (change in energy)/(change in time). Problem 10. A copper-bottomed kettle, with a bottom diameter of 24 cm and a thickness of 3.0 mm, sits on a burner. The kettle holds boiling water, and energy flows into the water from the kettle bottom at 800 W . What is the temperature of the bottom surface of the kettle? (Note: use k = 400 W/(m K) for the thermal conductivity of copper.) Problem 11. During heavy exercise, the body pumps 2.00 L of blood per minute to the surface, where it is cooled by 2.00 o C. What is the rate of heat transfer from this forced convection alone, assuming blood has the same specific heat capacity as water and its density is 1050 kg/m3 ? Problem 12. The glowing filament in a lamp is radiating energy at a rate of 60 W . At the filament’s temperature of 1500 o C, the emissivity is 0.23. What is the surface area of the filament? 3 Bonus Problems Problem Bonus 1. Your textbook (College Physics by OpenStax) defines (in equation 13.55) the average kinetic energy, KE (which, in class I have been calling Kavg ), of a particle1 in an ideal gas as 1 3 KE = m(v 2 ) = kB T , 2 2 where m is the mass of the particles that make up the gas (they are all assumed to have the same mass), (v 2 ) is the average square velocity of the particles in the gas (which is not the same as the average velocity squared), kB is Boltzmann’s constant, and T is the absolute temperature (in Kelvin). Your textbook calls this thermal energy. Technically speaking,  however, this equation has units of Joules/particle since we averaged the kinetic v 2 +v 2 +···+v 2 N energies (v 2 ) = 1 2 N . If we are going to call a quantity thermal energy it needs to be an energy and thus should have units of energy (Joules) - not units of Joules/particle (which are the units of a kind of quantity you might call an ”energy density”). Therefore, a better definition of thermal energy (which we will call Eth ) would be given by taking the above equation and multiplying it by the number of particles in the ideal gas N :  1 3 Eth = N KE = N m(v 2 ) = N kB T . 2 2 This is the thermal energy of an ideal gas of N particles. Note that this does indeed have units of energy (Joules). (This is the definition of thermal energy given by most standard textbooks - including the supplementary textbook, College Physics by Knight, Jones, and Field.) Now, suppose we have an ideal gas at 20 o C which consists of 2.2 × 1022 atoms. 4.3 J of energy are removed from the gas. What is the new temperature in o C? Problem Bonus 2. What is the greatest possible rate of energy transfer by radiation for a 5.0 cm-diameter sphere that is at 100 o C? (Recall that the surface area of a sphere is 4πr2 .) Hint: Think carefully about what value of the emissivity e will give the maximum possible rate of transfer of radiant heat. 1 Recall from class that I am using the words ”particle”, ”atom”, and ”molecule” interchangeably here.
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