College Physics I: BIIG problem-solving method

Created Feb. 7, 2024 by C N Hiremath

Temperature Scales

  • Thermometers are used to measure temperature according to well-defined scales of measurement, which use pre-defined reference points to help compare quantities.
  • The three most common temperature scales are the: Fahrenheit, Celsius, and Kelvin scales.

 

Celsius Scale

  • Anders Celsius, Swedish astronomer, invented the temperature scale in 1742.

Used freezing (0) and boiling points (100) of pure water.

The units are “degrees Celsius” or ˚C.

  • Fahrenheit scale is related to the Celsius scale by

            TC   =   ( 5 / 9 )  [ TF - 32˚ ]

 

Ideal Gas

  • Ideal gas atoms do not interact with each other (except for collisions)
  • The system has no potential energy.
  • The only internal energy in the ideal gas is the translational kinetic energy of the individual atoms.

 

Absolute zero

  • Fixed to the point at which kinetic energy of atoms is zero.
  • Kinetic energy is always positive, so the zero on the temperature scale will be an absolute zero
  • No temperature below this is possible.

 

Kelvin Scale

  • Zero degrees is the point at which the kinetic energy of atoms is zero.
  • All temperatures on the Kelvin scale are positive, so it is often called an absolute temperature scale.
  • The units of the Kelvin temperature scale are “kelvin” (not degrees kelvin) or K
  • The spacing between divisions for the Kelvin scale is the same as that of the Celsius scale

            ∆T (K)  =   ∆T (°C) 

  • The only difference is the position of the zero point.

            T   =   TC  +  273.15

  • Problem (E13.1):  “Room temperature” is generally defined to be 77ºF. What is it in K?            ( 300 K )

 

The Zeroth Law of Thermodynamics

  • If two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C.

 

Linear Thermal Expansion

  • The change in length ΔL is proportional to length L.
  • The dependence of thermal expansion on temperature, substance, and length is

            ΔL   =   α L ΔT

where,  ΔL is the change in length L,  ΔT is the change in temperature, and α is the coefficient of linear expansion, which varies slightly with temperature.

  • Because the size of a kelvin and a degree Celsius are the same, both α and ΔT can be expressed in units of kelvins or degrees Celsius.
  • Problem (E13.3):  The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest. The bridge is exposed to temperatures ranging from –15ºC to 40.ºC. What is its change in length between these temperatures? Assume that the bridge is made entirely of steel.               ( 0.84 m )

 

Thermal Expansion in Three Dimensions

  • The change in volume ΔV is

            ΔV   =   β V ΔT

where,  β is the coefficient of volume expansion, V is the volume V,  ΔT is the change in temperature.

  • Note that the values of β are almost exactly equal to 3α.
  • Problem (E13.4):  Suppose your 60.0-L (15.9-gal) steel gasoline tank is full of gas, so both the tank and the gasoline have a temperature of 15.0ºC. How much gasoline has spilled by the time they warm to 35.0ºC?                                                                                                                                              ( 1.10 L )

 

Ideal Gas Law

  • The ideal gas law states that

            P V   =   N k T

where,  P is the absolute pressure of a gas, V is the volume it occupies, N is the number of atoms and molecules in the gas, and T is its absolute temperature, k is the constant called the Boltzmann constant in honor of Austrian physicist Ludwig Boltzmann (1844–1906) and has the value

            k   =   1.38 × 10−23 J/K.

  • STP:  The conditions, 1 atm pressure and 0˚C, are called standard temperature and pressure or STP.
  • Problem (E13.6):  Suppose your bicycle tire is fully inflated, with an absolute pressure of 7.00 × 105 Pa (a gauge pressure of just under 90.0 lb/in2) at a temperature of 18.0ºC. What is the pressure after its temperature has risen to 35.0ºC? Assume that there are no appreciable leaks or changes in volume.                                                                                                                                             ( 7.41 × 105  Pa )

Moles and Avogadro’s Number

  • A mole (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12.
  • The actual number of atoms or molecules in one mole is called Avogadro’s number (NA).

NA   =   6.02 × 1023 mol−1

The number is independent of the type of gas.

  • Problem (E13.CYU):  The active ingredient in a Tylenol pill is 325 mg of acetaminophen

(C8 H9 NO2). Find the number of active molecules of acetaminophen in a single pill.              ( 1.30 × 1021 ) 

 

Ideal Gas Law (in terms of moles)

  • The ideal gas law (in terms of moles) is

                  P V   =   n R T

  • The numerical value of R in SI units is

            R   =   NA k   =   ( 6.02 × 1023 mol−1  )  ( 1.38 × 10−23 J/K )   =   8.31  J/mol  K

This is universal gas constant.

  • The ideal gas law can be considered to be another manifestation of the law of conservation of energy.

 

Kinetic Theory

  • The explanation of properties of the gas as a whole in terms of randomly moving particles is called kinetic theory.
  • Pressure: The quantity force-to-area ratio F/A is called the gas pressure.

            P   =   F / A

The pressure has units of N/m2. The SI unit of pressure is the pascal, defined as

1 pascal   =   1 Pa   =   1 N/m2

  • Temperature:  Temperature measures the average kinetic energy of the particles in a system.
  • rms speed:  The root-mean-square speed of ideal gas particles is

            vrms   =   √ ( v 2 )avg

  • Average kinetic energy:  The average kinetic energy of the gas particles is

            Kave   =   ½ m ( v 2 )avg    =   ½ m vrms2

  • The average kinetic energy of the gas particles is directly proportional to the absolute temperature T

Kave   =   ( 3/2 )  kB T

where, kB is the boltzmann’s constant.

 

Temperature and Kinetic Energy

  • Temperature on the Kelvin scale of an ideal gas is related to the average kinetic energy per atom by

            T   =   2/3 ( Kavg / kB )

where, kB  is a constant known as Boltzmann’s constant.

            kB   =   1.38 x 10-23 J/K

  • The root-mean-square speed of the atoms in a gas is

vrms   =   √ ( 3 kB T / m)

  • Problem (E13.10):  Nitrogen gas consists of molecules, N2.  At 20.0˚C, what is the root-mean-square speed of the molecules of nitrogen? Given that the atomic mass (u) of nitrogen is 14.003074 g.   

      ( 511 m/s )  

 

Maxwell-Boltzmann Distribution

  • The motion of molecules in a gas is random in magnitude and direction for individual molecules,

but a gas of many molecules has a predictable distribution of molecular speeds.

  • This distribution is called the Maxwell-Boltzmann distribution,after its originators, who calculated it based on kinetic theory, and has since been confirmed experimentally.
  • The distribution has a long tail, because a few molecules may go several times the rms speed.
  • The curve is shifted to higher speeds at higher temperatures, with a broader range of speeds.

 

Phase Diagrams

  • The plots of pressure versus temperatures provide considerable insight into thermal properties of substances. There are well-defined regions on these graphs that correspond to various phases of matter, so PT graphs are called phase diagrams.
  • Critical point:  There is a critical point—that is, a critical temperature—above which liquid cannot exist.
  • Critical pressure is the minimum pressure needed for liquid to exist at the critical temperature.
  • Triple point:  All three curves on the phase diagram meet at a single point, the triple point, where all three phases exist in equilibrium. For water, the triple point occurs at 273.16 K (0.01ºC).


BIIG: Problems & Solutions


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