Consider two boxes containing ideal gases A and B such that their temperatures, pressures and number densities are same. The molecular size of A is half of that of B and mass of molecule A is four times that of B. If the collision frequency in gas B is 32 × 10¹⁸/s$32 \times 10^{18}/\mathrm{s}$ then collision frequency in gas A is ____/s.

Related Formula

Core Logic

The parameter asked is the mean squared velocity, which corresponds directly to Vrms²$\mathrm{V}_{\mathrm{rms}}^2$.

From the ideal gas kinematics relation, we observe:

Comparing this format against standard geometric linear templates (y = mx$y = mx$), the curve must map as a clean straight line originating from absolute zero zero coordinates.

Step 1: Final Conclusion

This linear profile matches Graph (1), selecting option (1).

Pattern Recognition

Watch the vertical ordinate labels carefully: Root-mean-square velocity scales as a sub-linear curve (T$\sqrt{\mathrm{T}}$), while mean squared metric trends linearly directly (y ∝ x$y \propto x$).

Chapter Mix

Class 11 Physics: Kinetic Theory

Related Formula

Ideal Gas Law:

Conservation of moles: n₁ + n₂ = nf$n_1 + n_2 = n_f$

Core Logic

Let the volume of the smaller vessel be V₁ = V$V_1 = V$, then the volume of the larger vessel is V₂ = 2V$V_2 = 2V$. Initial moles in large vessel:

Initial moles in small vessel:

Total total initial moles:

Step 1: Connect Vessels to Dynamic Equilibrium

When connected, the total final volume is Vf = V + 2V = 3V$V_f = V + 2V = 3V$. The final temperature is Tf = 600 K$T_f = 600\text{ K}$. Using mole conservation:

Pf = (30 × 200)/(1000) = 6 kPa$$P_f = \frac{30 \times 200}{1000} = 6\text{ kPa}$$

Pattern Recognition

Connecting chambers preserves the net mass/moles (Σ nᵢ = constant$\sum n_i = \text{constant}$). Keep everything relative to a common volume multiplier V$V$ to easily cancel terms.

Chapter Mix

Class 11 Physics: Kinetic Theory

Related Formula

where:

• f$f$ = frequency of collisions
• vavg$v_{\text{avg}}$ = average speed of the molecules
• λ$\lambda$ = mean free path

Core Logic

Given parameters:

• Average speed, vavg = 600~m/s$v_{\text{avg}} = 600\mathrm{~m/s}$
• Mean free path, λ = 3 × 10⁻⁷~m$\lambda = 3 \times 10^{-7}\mathrm{~m}$

Step 1: Calculate Frequency

Substitute the values into the formula:

Hence, the collision frequency is 2 × 10⁹/s$2 \times 10^{9}/\mathrm{s}$.

Pattern Recognition

Collision frequency is simply distance covered per unit time (average velocity) divided by the average distance between consecutive collisions (mean free path).

Chapter Mix

Class 11 Physics: Kinetic Theory

Related Formula

K.E. = (f)/(2) kB T$$\text{K.E.} = \frac{f}{2} k_B T$$ [cite: 688]

Core Logic

The average kinetic energy per molecule depends only on the temperature T$T$ and the degrees of freedom f$f$ of the gas[cite: 75, 688]. Both Helium (He$\mathrm{He}$) and Argon (Ar$\mathrm{Ar}$) are monoatomic noble gases, meaning both share the same degrees of freedom (f = 3$f = 3$)[cite: 689]. Since they sit in the same flask at identical room temperature (T = 300 K$T = 300\ \text{K}$), their translational kinetic energies per molecule are exactly equal [cite: 74, 688]:

K.E.HeK.E.Ar = (1)/(1)$$\frac{\text{K.E.}_{\mathrm{He}}}{\text{K.E.}_{\mathrm{Ar}}} = \frac{1}{1}$$ [cite: 688]

Pattern Recognition

Do not get distracted by the molar masses given in the question stem[cite: 76, 77]. Kinetic energy per molecule is purely temperature-dependent for an ideal gas, unlike the root-mean-square velocity (vrms$v_{\text{rms}}$) which explicitly includes molecular weight.

Chapter Mix

Class 11 Physics: Kinetic Theory of Gases

Related Formula

The root-mean-square (r.m.s.) velocity of gas molecules is given by:

Since molecular weight M$M$ is directly proportional to the vapour density (ρ$\rho$), the r.m.s. velocity is inversely proportional to the square root of its vapour density:

Core Logic

Given the ratio of vapour densities :

Therefore, the ratio of their r.m.s. velocities is:

Pattern Recognition

R.M.S. velocity changes inversely with the square root of mass or density. Whenever a density ratio is given, simply invert the fraction and take the square root to immediately find the velocity ratio.

Chapter Mix

Class 11 Physics: Kinetic Theory of Gases