There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at 8 kPa at 1000 K while the smaller vessel contains the gas at 7 kPa at 500 K. If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at 600 K, at steady state the pressure in the vessels will be (in kPa).

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

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