Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

JEE: Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The document Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE is a part of the JEE Course I. E. Irodov Solutions for Physics Class 11 & Class 12.
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Q. 220. Calculate what fraction of gas molecules
 (a) traverses without collisions the distances exceeding the mean free path λ;
 (b) has the free path values lying within the interval from λ to 2λ.

Solution. 220. (a) The fraction of gas molecules which traverses distances exceeding the mean free path without collision is just the probability to traverse the distance s = λ without collision.

Thus  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) This probability is   Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 221. A, narrow molecular beam makes its way into a vessel filled with gas under low pressure. Find the mean free path of molecules if the beam intensity decreases η-fold over the distance Δl. 

Solution. 221. From the formula 

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 222. Let αdt be the probability of a gas molecule experiencing a collision during the time interval dt; α is a constant. Find:
 (a) the probability of a molecule experiencing no collisions during the time interval t; 
 (b) the mean time interval between successive collisions.

Solution. 222. (a) Let P (f) = probability of no collision in the interval (0, t). Then

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

where we have used P (0) = 1

(b) The mean interval between collision is also the mean interval of no collision. Then

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 223. Find the mean free path and the mean time interval be- tween successive collisions of gaseous nitrogen molecules
 (a) under standard conditions;
 (b) at temperature t = 0°C and pressure p = 1.0 nPa (such a pressure can be reached by means of contemporary vacuum pumps). 

Solution. 223.

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE
Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 224. How many times does the mean free path of nitrogen molecules exceed the mean distance between the molecules under standard conditions? 

Solution. 224. The mean distance between molecules is of the order

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

This is about 18.5 times smaller than the mean free path calculated in 2.223 (a) above.


Q. 225. Find the mean free path of gas molecules under standard conditions if the Van der Waals constant of this gas is equal to b = 40 ml/mol. 

Solution. 225. We know that the Vander Waal’s constant b is four times the molecular volume. Thus

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Hence  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 226. An acoustic wave propagates through nitrogen under standard conditions. At what frequency will the wavelength be equal to the mean free path of the gas molecules?

Solution. 226. The volodty of sound in N2 is

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

so,  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or,   Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 227. Oxygen is enclosed at the temperature 0°C in a vessel with the characteristic dimension l = 10 mm (this is the linear dimension determining the character of a physical process in question). Find: 

(a) the gas pressure below which the mean free path of the molecules λ > 1;
 (b) the corresponding molecular concentration and the mean distance between the molecules. 

Solution. 227. 

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Now   Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) The corresponding n is obtained by dividing by kT and is  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE m3 = 1.84 uper c.c. and the corresponding mean distance is Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 228. For the case of nitrogen under standard conditions find:
 (a) the mean number of collisions experienced by each molecule per second;
 (b) the total number of collisions occurring between the molecules within 1 cm3 of nitrogen per second.

Solution. 228. 

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) Total number of collisions is

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Note, the factor 1/2. When two molecules collide we must not count it twice.


Q. 229. How does the mean free path λ and the number of collisions of each molecule per unit time v depend on the absolute temperature of an ideal gas undergoing 

(a) an isochoric process;
 (b) an isobaric process?

Solution. 229. 

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

d is a constant and n is a constant for an isochoric process so X is constant for an isochoric process.

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE for an isobaric process.

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE or an isobaric process.


Q. 230. As a result of some process the pressure of an ideal gas increases n-fold. How many times have the mean free path λ and the number of collisions 'of each molecule per unit time v changed and how, if the process is
 (a) isochoric;
 (b) isothermal?

Solution. 230. (a) In an isochoric process λ is constant and

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE must decrease n times in an isothermal process and v must increase n times because <v > is constant in an isothermal process.


Q. 231. An ideal gas consisting of rigid diatomic molecules goes through an adiabatic process. How do the mean free path λ and the number of collisions of each molecule per second v depend in this process on
 (a) the volume V;
 (b) the pressure p;
 (c) the temperature T? 

Solution. 231. 

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

But In an adiabatic  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE
or  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus    Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(c)  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

But   Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 232. An ideal gas goes through a polytropic process with exponent n. Find the mean free path and the number of collisions of each molecule per second v as a function of
 (a) the volume V;
 (b) the pressure p;
 (c) the temperature T. 

Solution. 232. In the polytropic process of index n

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

so  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 233. Determine the molar heat capacity of a polytropic process through which an ideal gas consisting of rigid diatomic molecules goes and in which the number of collisions between the molecules remains constant
 (a) in a unit volume;
 (b) in the total volume of the gas. 

Solution. 233. (a) The number of collisions between the jnolecules in a unit volume is

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

This remains constant in the poly process pV-3 = constant

Using (Q.122) the molar specific heat for the polytropic process

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

is Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

It can also be written as  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) In this case  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

so   Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

It can also be written as  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 234. An ideal gas of molar mass M is enclosed in a vessel of volume V whose thin walls are kept at a constant temperature T. At a moment t = 0 a small hole of area S is opened, and the gas starts escaping into vacuum. Find the gas concentration n as a function of time t if at the initial moment n (0) = n0.

Solution. 234. We can assume that all molecules, incident on the hole, leak out. Then,

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or   Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Integrating  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 235. A vessel filled with gas is divided into two equal parts 1 and 2 by a thin heat-insulating partition with two holes. One hole has a small diameter, and the other has a very large diameter (in comparison with the mean free path of molecules). In part 2 the gas is kept at a temperature η  times higher than that of part 1. How will the concentration of molecules in part 2 change and how many times after the large hole is closed? 

Solution. 235. If the temperature of the compartment 2 is η times more than that of compartment 1, it must contain 1/η times less number of molecules since pressure must be the same when the big hole is open. If M - mass of the gas in 1 than the mass of the gas in 2 must be M/η . So immediately after the big hole is closed.

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

where m = mass of each molecule and  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE are concentrations in 1 and 2. A fter the big hole is closed the pressures will differ and concentration will become n1 and n2 where

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

On the other hand

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

So Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 236. As a result of a certain process the viscosity coefficient of an ideal gas increases α = 2.0 times and its diffusion coefficient β = 4.0 times. How does the gas pressure change and how many times? 

Solution. 236. We know

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus η changing α times implies T changing α2 times. On the other hand

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus D changing P times means Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE changing β times

So p must change  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 237. How will a diffusion coefficient D and the viscosity coefficient η of an ideal gas change if its volume increases n times: (a) isothermally; (b) isobarically? 

Solution. 237. Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(a) D will increase n times

η will remain constant if T is constant

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus D will increase   Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE if p is constant


Q. 238. An ideal gas consists of rigid diatomic molecules. How will a diffusion coefficient D and viscosity coefficient η change and how many times if the gas volume is decreased adiabatically n =10 times? 

Solution. 238. 

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

In an adiabatic process

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Now V is decreased  Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

So D decreases n4/5 times and η increase n1/5 times.

The document Irodov Solutions: Transport Phenomena- 1 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE is a part of the JEE Course I. E. Irodov Solutions for Physics Class 11 & Class 12.
All you need of JEE at this link: JEE

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