IIT JEE / CET – Main / Advanced Physics 2013 & 2014 : Physics Free notes – Thermodynamics,Kinetic theory – All the essentials in one page [ PO’PS.Series.1 ]

All prevailing thermodynamic physics - from the stars to the wave surfer

T H E R M O D Y N A M I C S

The branch of physics which deals with the relationships among heat , work , energy and temperature . Any physical system will spontaneously reach an equilibrium that can be described by specifying its properties, such as pressure , temperature, or chemical composition . Eventhough it’s based on physics , this branch has applications across all scientific streams including life sciences . In fact , physical life itself can be described as a continual thermodynamic cycle  of transformations between heat and energy. But these transformations are never perfectly efficient .

Def  : The branch of physics which deals with the laws that control  –

  • the conversion of energy from one form to another
  • the direction of the energy / heat flow
  • the availability and other parameters required for energy to do work.

It works on the premise there should be a measurable quantity of energy called the internal energy ( U ) in an isolated system , anywhere in the universe . Internal energy here refers to the total kinetic and potential energy of the atoms and molecules of the system of any kind which can be transferred directly in the form of heat – hence , it excludes chemical and nuclear energy .

Internal energy ( U ) changes , when  –

  • the system is not isolated   i.e,
  • there is transfer of mass to or from the system  or
  • there is transfer of heat ( Q ) to or from the system or
  • there is work ( W ) being done on or by the system.

KINETIC THEORY OF GASES

Animation : The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms. Shown here is the movement of helium atoms .

Kinetic theory of gases was basically devised to develop a model of the molecular behaviour which could account for the observed behaviour of an ideal gas.It deals with both the macroscopic ( like temperature , pressure ) as well as the microscopic ( like kinetic energy of molecules , momentum , speed ) properties of gas molecules .Using the kinetic theory, scientists can relate the independent motion of molecules of gases to their volume, viscosity, and heat conductivity.

Assumptions  –

  1. Every gas contains extremely small particles called molecules which are all identical , but different from that of another gas.
  2. Molecules of a gas are identical , spherical , rigid and perfectly elastic point masses.
  3. Their molecular size is negligible when compared to intermolecular distances.
  4. The volume of molecules is negligible when compared to the volume of a gas .
  5. In a gas , molecules move in all possible directions with all possible speeds in accordance with Maxwell’s distribution law.
  6. The gas molecules undergo perfectly elastic collisions with themselves and with the walls of the vessel.
  7. The number of molecules is so large that a statistical treatment can be applied.
  8. Collisions are characterized by molecules moving in straight lines with constant speeds.
  9. Free path is the distance covered by the molecules between two sucessive collisions . The mean of all free paths is called a mean free path.
  10. Except during collisions the interactions among molecules are negligible , i.e, no forces are exerted on one another.
  11. The time taken for a molecular collision is negligible when compared to the time between two sucessive collisions.
  12. For a gas , the no of collisions / unit volume remains constant.
  13. Due to the very small masses and very high speeds of the molecules , gravitational attraction between them is ineffective.
  14. Change in momentum caused by the molecules striking the container walls is transferred to the walls.Due to this the gas molecules exerts a pressure on the walls of the container.
  15. Density of a gas is constant at all points of the container.
  16. The average kinetic energy of the gas molecules depends only on the temperature of the system.
  17. Relativistic and quantum – mechanical effects are negligible.
  18. The equations of motion of the molecules of an ideal gas are time-reversible.

Ideal gas pressure

Let us consider an ideal gas containing N molecules , each of mass m . It’s confined to a box of sides a, b and c . Now , consider a molecule which hits the wall of the box with velocity v1. After hitting , v1 gets resolved into components vx,vy and vz along x , y and z axis respectively. The change in momentum along the x axis due to the collision is

Δ p = mvx – ( -mvx) = 2mvx .

Now , the time interval between succesive collisions on the same side a is 2a/v . The no of collisions per second by a single molecule on the same side is vx/2a. Hence , the change in momentum / second on the wall due to a single collision of a single molecule is –

vx/2a  x  2mvx =  mvx²/a  . Therefore , the change in momentum for all the collisions on a particular wall will be  ∑ ( mvx²/a )

Now , rate of change of momentum = force [ Newton’s second law ]. Hence , Fx =  ∑ ( mvx²/a )   =   m/a ∑ vx² .  Also , pressure = force/unit area , hence  Px = F/A   =   Fx/bc    =    m/abc ∑  vx²      =     m/V  ∑ vx²  . This means that ,

Px + Py + Pz  =    m/V   ∑ ( vx² +vy² + vz² )  . Now , as P = Px+Py + Pz   and  v²  = vx² +vy² + vz²   , hence   –

3 P = m/V  ∑  v²   . Now , the mean square velocity  =  (|v|)²  =   ( v1²  + v2² + ……. )/N  =   ∑ v² / N

Hence ,  3 P  = (m/V)  x N x  (|v|)²     PV = 1/3 mN  (|v|)²     therefore ,

P  =  1/3   mN/V (|v|)²

Points to note  –

  • Pressure,volume , mass and temperature are related by –

P  =  1/3 ( mN/V )  v²rms    i.e, [ as v²rms α T ]     P  α ( MN ) T/V

  • When V and T are constant ,   P  α m N i.e, pressure α mass of the gas . Hence, when the mass of a gas increases , due to the increase in the no of molecules , the no of collisions / second also increases , leading to increased gas pressure.
  • When m and T are constant , P α v²rms α T . This means that as the temperature increases , the mean square speed of the molecules also increases, and hence the rate of collision between themselves and with the walls of the container also increases.

P = 1/3 ( mN/V )  v²rms   . Now , as M = mN = total mass of the gas ,

P = 1/3 ( M/V )  v²rms    =    1/3 ρ v²rms –     eq.1

  • For an ideal gas , pressure and kinetic energy are related as  –

K.E = 1/2 M v²rms   =   1/2 [ M/V ] v²rms  ( for unit volume )  =  1/2 ρ v²rms   – eq.2

Combining eqs 1 & 2 ,   P = 2/3 E

  • Hence , pressure of an ideal gas is numerically equal to two-thirds of the mean kinetic energy of translation ( kinetic energy due to motion from one location to another ) per unit volume of the gas.

IDEAL  GAS  LAW

An ideal / perfect gas refers to a gas which obeys the ideal gas law. Eventhough real gases doesn’t obey ideal gas law, at extremely low pressures and high temperatures , some gases like nitrogen ,helium and hydrogen behave quite like ideal gases.

The standard form of this law relates to the pressure ( P ) , volume (V) and temperature ( T ) of an ideal gas  –

PV = nRT where n is the amount of gas and R is the Regnault constant or Universal gas constant .  Some variations are –

  • For 1 mole / M grams / 22.4 L of gas  – PV = RT
  • For one  molecules of gas     –            PV = ( R/NA) T = kT
  • For N molecules of gas     –                 PV = NkT
  • For μ moles of gas      –                      PV =  μRT
  • For 1 gm of gas     –                          PV = ( R/M) T = rT

UNIVERSAL / MOLAR / IDEAL GAS CONSTANT  ( R ) [ Dimensions – M L2 T -2 θ -1 ]

  • It’s a physical constant present in a large number of fundamental equations in the physical sciences, like the ideal gas equation and the Nernst equation. It’s equivalent to the Boltzmann constant , but is expressed in units of energy per kelvin per mole.

R  = PV/μ T   = Pressure x volume / no of moles x temperature = Work done / no of moles x temperature

STP values  =  8.32 joule /mole x kelvin  =  1.98  cal / mole x kelvin  = 0.8221  litre x atmos / mole x kelvin

BOLTZMANN’S  CONSTANT  ( k ) [ Dimensions –  M L2 T -2 θ -1 ]

  • k  = R/N  =  8.31 / 6.023 x 1o 23 =  1.38 x 10 -23 J / K

SPECIFIC GAS CONSTANT  ( r ) [ Dimensions –  L2 T -2 θ -1 ]

  • r = R / M    As the value of M is unique to a gas , r will be different for different gases .
  • r can also be calculated as  r = Cp – Cv , where  Cp is the specific heat for a constant pressure and Cv  is the specific heat for a constant volume.

GAS   MOLECULE   SPEEDS

Root mean square speed –  defined as the square root of mean of squares of the speed of the molecules in a gas

  • vrms = √ ( v1² + v2² + v3² + v4² ….. ) / N   .

Now , P = 1/3 ( mN/V )  v²rms hence,         v²rms = √ 3 P V/mN

√ 3 P V/mass of gas     =     √ 3 P/ρ [ from ρ = gass mass / V ]

  • vrms = √  3PV/mass of gas    =   √3μRT/μM  =   √ 3 RT/M

=  √ 3 NAk T / NAM    =    √ 3 k T/m

Points  to  note  –

  • As  vrms 1/α √M , more the molecular weight of the gas molecules , lesser their root mean square speeds.
  • As  vrms α √ T , more the temperature of the gas molecules , greater is their rms speeds. Hence , when T = 0 , vrms = 0, this temperature at 0 K is referred to as the absolute zero.
  • As the vrms of gas molecules is greater than the escape velocity of moon , moon has no atmosphere.
  • At constant temperature , vrms speed of gas molecules doesn’t change if pressure is increased ,i.e. if pressure is increased n times, eventhough the density will increase by n times ,  vrms speed will remain the same .
  • The vrms speed of gas molecules is expressed in kms / second.

Most probable speed

Defined as the speed with which the maximum fraction of the total number of molecules of a gas moves .

vmp =   √ 2 P/ρ =    √ 2 RT/M =   √ 2 k T/m

Average  speed    –

Defined as the arithmetic mean of the speed of molecules of a gas at a given temperature .

vav =  (  v1 + v2 + v3 + v4 + …………….   ) / N

vav =    √ 8 P/π ρ =      √ 8 RT/π M =   √ 8 k T/π m

Note   –

  • vrms >  vav >  vmp and    vrms : vav :  vmp =      √ 3  :  √2.5  :  √2

(  to continue …………. )

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