Tag: law of equipartition of energy and mean free path

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Multiple choice physics behaviour of perfect gas and kinetic theory of gases mean free path law of equipartition of energy and mean free path behavior of perfect gas and kinetic theory

There are two vessels of same consisting same no of moles of two different gases at same temperature . One of the gas is $CH _{4}$ & the other is unknown X. Assuming that all the molecules of X are under random motion whereas in $CH _{4}$ except one all are stationary. Calculate $Z _{1}$ for X in terms of $Z _{1}$ of $CH _{4}$. Given that the collision diameter for both gases are same & $\displaystyle (U _{rms}) _{x}=\frac{1}{\sqrt{6}}(Uav) _{CH _{4}}$.

  1. $\displaystyle \frac{2\sqrt{2}}{3\sqrt{\pi }}Z _{1}$

  2. $\displaystyle \frac{3\sqrt{2}}{2\sqrt{\pi }}Z _{1}$

  3. $\displaystyle \frac{2\sqrt{3}}{2\sqrt{\pi }}Z _{1}$

  4. $\displaystyle \frac{4\sqrt{2}}{3\sqrt{\pi }}Z _{1}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

V, n, T $\rightarrow  same$(25) so $P\rightarrow $ also same ( P  5  25)
$\displaystyle \sigma \rightarrow same (25)$
given

$\displaystyle (v {rms})\times

x=\dfrac{1}{\sqrt{6}}(v _{avg.}) _{CH _{4}}$ &

$v _{rms}=\sqrt{\dfrac{3\pi }{8}}(v _{avg.})$ so
$\displaystyle \sqrt{\dfrac{3\pi }{8}}(v _{avg.}) _{CH _{4}}$
$\displaystyle \dfrac{(v _{avg.})x}{(v _{avg.}) _CH _{4}}=\sqrt{\dfrac{8}{3\pi }}.\frac{1}{\sqrt{6}}=\dfrac{2}{3\sqrt{\pi }}$
For X (9< ) : $\displaystyle Z _{1}=\sqrt{2}\pi \sigma ^{2}(v _{avg.}) _{x}N^{\ast }$
For CH{4} (9< ) : $\displaystyle Z _{1}=\pi \sigma ^{2}(v _{avg.}) _{CH _{4}}N^{\ast }$
Since T, P, v, n are same, $N\ast $ will also be same.
$\displaystyle



\frac{Z _{1}X}{Z _{1}(CH _{4})}=\sqrt{2}\frac{(v _{avg.}) _{x}}{(v _{avg.}) _{CH _{4}}}=\sqrt{2}.\frac{2}{3\sqrt{\pi

}}$
$\displaystyle Z _{1}(X)=Z _{1}(CH _{4}).\frac{2\sqrt{2}}{3\sqrt{\pi }}$

Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

How many degrees of freedom the gas molecules have if under STP the gas density $\rho = 1.3 kg/m^3$ and the velocity of sound propagation in it is $330 ms^{-1}$?

  1. $3$

  2. $5$

  3. $7$

  4. $8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using v = sqrt(gamma * P / rho) and P = (rho/M)RT, we find gamma. Since v = 330 m/s, rho = 1.3 kg/m^3, and P = 1.013e5 Pa, gamma = v^2 * rho / P = 1.71. However, for standard gases, gamma = 1 + 2/f. With gamma = 1.4 (diatomic), f = 5. The provided data yields gamma approx 1.4, corresponding to f = 5.

Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

At room temperature (27$^0$ C) the rms speed of the moleculesof certain diatomic gas is found to be 1920 ms$^{-1}$ then the molecule is:

  1. $H _2$

  2. $F _2$

  3. $O _2$

  4. $Cl _2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Let the room temperature is $T = 27^0C=27+273=300K$
Now, $V _{rms}=\sqrt{\dfrac{3RT}{m}}$
$\Rightarrow M=\dfrac{3RT}{V _{rms}^2}$
By putting the value we get,
$M=\dfrac{3\times8.314\times300}{1920^2}=2\times10^{-3}kg=2g$
Thus, it is an Hydrogen.
Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

If temperature of body increases by 10%, then increase in radiated energy of the body is :

  1. 10 %

  2. 40 %

  3. 46 %

  4. 1000 %
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to the Stefan-Boltzmann law, radiated energy E is proportional to T^4. If T increases by 10%, T_new = 1.1T. Then E_new = (1.1)^4 * E = 1.4641 * E. The increase is 46.41%.

Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

The law of equipartition of energy was given by :

  1. Claussius

  2. Maxwell

  3. Boltzmann

  4. Carnot
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The law is given by Claussius which states that for any dynamical system in a thermal equilibrium, the total energy is equally divided among the degrees of freedom.

Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

The value of $\gamma$ for gas X is 1.66, then x is :

  1. Ne

  2. O$ _3$

  3. N$ _2$

  4. H$ _2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given that value of gamma is 1.66 i.e $\dfrac{5}{3}$ which implies that it is a monoatomic gas, and Neon (Ne) is the only monoatomic gas among the given options.

Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

An ant is moving on a plane horizontal surface. The number of degrees of freedom of the ant will be

  1. 1

  2. 2

  3. 3

  4. 6
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Degrees of freedom represent the number of independent coordinates required to specify the position of an object. An ant on a 2D plane requires two coordinates (x, y).

Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

Gas exerts pressure on the walls of container because the molecules-

  1. Are loosing the kinetic energy

  2. Are getting stuck to the walls

  3. Are transferring their momentum to walls

  4. Are accelerated toward walls.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Gas molecules are in random motion having some momentum and while colliding with the walls they transfer their momentum to the walls and this collective transfer of momentum from all the molecules to the walls appears as pressure exerted by gas on the container wall.

Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

A man is climbing up a spiral type staircase. His degrees of freedom are :

  1. 1

  2. 2

  3. 3

  4. more than 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

There will be three degrees of freedom. Two are along x-direction and y-directions due to translation and the last degree of freedom due to angular rotation as the  man climbs up.

Hence, Option C is correct.

Multiple choice physics behaviour of perfect gas and kinetic theory of gases degree of freedom: law of equipartition of energy law of equipartition of energy law of equipartition of energy and mean free path

A system consists of N particles, which have independent K relations among one another. The number of degrees of freedom of the system is given by :

  1. 3 NK

  2. N/3K

  3. 3 N/K

  4. 3N - K
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a system of N particles having K independent relations among them, the degrees of freedom of the system is given by 3N-K. 3N  is due to three degrees of freedom associated with each particle if all the particles are independent of each other (i.e K=0) and due to K relation among them, degrees of freedom reduces to 3N-K

Hence, Option D is correct.