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Test: Specific Heat Capacity & Mean Free Path (September 22) - NEET MCQ


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10 Questions MCQ Test - Test: Specific Heat Capacity & Mean Free Path (September 22)

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Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 1

The mean free path of moving gas molecules is directly proportional to kth power of diameter of molecule. Here value of k is

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 1

The expression for the mean free path is given by:

Here, the area of the cross-section is πd2.

The volume of the cylindrical path is given by πd2 x vt, where t is the time, and v is the molecule’s velocity and the number of molecules per unit volume as (N/V).

Thus we can say that;

λ ∝ 1/d2 or d-2

∴ k = -2

Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 2

Calculate the value of mean free path (λ) for oxygen molecules at temperature 27°C and pressure 1.01 x 105 Pa. Assume the molecular diameter 0.3 nm and the gas is ideal. (k = 1.38 x 10-23 JK-1

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 2

Given : 
T = 27oC = 27 + 273.15 = 300.15 K
P = 1.01 x 105 Pa
d = 0.3 nm = 0.3 x 10-9
k = 1.38 x 10-23 JK-1
We know that PV = NkT then after rearranging we get,

Mean free path for oxygen molecules

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Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 3

For mean free path λ:

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 3

Given:

From the ideal gas equation

PV = nRT

where 'n' is the number of the moles

Here nNA is the number of molecules in n moles.

Here  is the number of molecules per unit volume of the gas and  is the Boltzmann  constant.

The mean free path of a gas molecule is given by:

∴ with an increase in the number of molecules per unit volume, the mean free path decreases.

∴ λ ∝ (1/d2) (Option 4 is not true)

∴ with a decrease in the size of molecules, the mean free path increases. (Option 3 is true)

At constant temperature T, the mean free path will decrease on increasing P. (Option 1 is not true)

Similarly, 

At constant pressure P, the mean free path will increase on increasing T. (Option 4 is not true)

Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 4

The mean free path of a gas molecule is λ1 at temperature T. The pressure and temperature, of both the gas, are made two times their original value. The new mean free path was found to be λ2. Then λ12 is -

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 4

From the ideal gas equation -

PV = nRT

Where n is the number of the moles

Here nNA is the number of molecules in n moles.

Here,  is the number of the molecules per unit volume of the gas and  = k, the Boltzmann constant

Here we can see that,
λ ∝ T and λ ∝ (1/P)
∴ On increasing T and P two times, the net effect is zero on λ.
Hence λ2 = λ1
⇒ λ12 = 1 : 1

Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 5

If the gas particles are of diameter 'd', average speed 'v', number of particles per unit volume 'n', then the time between two successive collisions on average is __________.

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 5

The volume swept by the molecules in small time Δt, in which molecules will collide with it is 

 

Where d = distance between the center of two molecules,  v = average speed of the molecules

  • If n is the number of molecules per unit volume of the gas, then the collision suffered by the molecule in time  Δt is πd2vΔt x n
  • The number of collisions per sec is
  • The average time between two successive collisions is 
Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 6

If the gas particles are of diameter 'd', average speed 'v', number of particles per unit volume 'n', then what is the term "Πd2vt" represents?

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 6

Suppose d is the diameter of each molecule of the gas.
A particular molecule will suffer a collision with any molecule that comes within a distance d between the centers of two molecules.
If  is the average speed of the molecules.
The volume swept by the molecules in small time Δt, in which molecules will collide with it is 

 

Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 7

The mean free path of a gas molecule depends on

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 7


From the above, it is clear that the mean free path of a gas molecule depends on the number of molecules per unit volume and diameter of a molecule. Therefore option 3 is correct.

Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 8

The mean free path of gas molecules is proportional to nth power of diameter of molecules. Here n is

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 8

The mean free path of a gas molecule is given by

Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 9

The mean free path λ of a gas molecule as given by Maxwell is related to its diameter a, as 
 

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 9

 is the mean free path given by Maxwell. So option 2 is correct.
In the given options, m is the mass of the molecules and T is the temperature of the gas.

Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 10

The relation between mean free path λ and the pressure P of any gas is - 

Detailed Solution for Test: Specific Heat Capacity & Mean Free Path (September 22) - Question 10

Mean Free Path (λ):

  • The distance traveled by a gas molecule between two successive collisions is known as a free path.
  • During two successive collisions, a molecule of gas moves in a straight line with constant velocity, and the mean free path of a gas molecule is given by

n is the number of molecules per unit volume.

Here KB is Boltzman Constant, P is pressure, T is temperature
Now if we put (2) in (1) we get

K is constant, T is the temperature which is constant for isothermal Gas.

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