Sigma Percentile
JEE Main 2019
LEVELJEE Advanced

Animated Solution for Physics - Kinetic Theory: The temperature, at which the root mean square velocity of hydrogen molecules equals their escape velocity from the earth, is closest to : [Boltzmann constant J/K, Avogadro number /kg, Radius of earth m, Gravitational acceleration on earth ]

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Visualized Solution

  • Let's visualize a hydrogen molecule on the surface of the Earth.
  • We need to find the temperature at which its root mean square speed equals the escape velocity .

  • The escape velocity from the surface of the Earth is given by:
  • v_e = \sqrt{2gR_e}

  • The root mean square velocity of a gas molecule at temperature is:
  • v_{rms} = \sqrt{\frac{3k_B T}{m}}
  • where is the mass of one molecule.

  • Equating the two velocities:
  • \sqrt{\frac{3k_B T}{m}} = \sqrt{2gR_e}
  • Squaring both sides:
  • \frac{3k_B T}{m} = 2gR_e

  • The mass of one molecule is its molar mass divided by Avogadro's number.
  • Notice the units of given: . This is the number of molecules in a kilomole!
  • Since the molar mass of is :
  • m = \frac{2}{N_A} \text{ kg}

  • Rearranging for :
  • T = \frac{2gR_e m}{3k_B}
  • Substitute :
  • T = \frac{2gR_e \times 2}{3k_B N_A}

  • Substitute the given values:
  • T = \frac{2 \times 10 \times 6.4 \times 10^6 \times 2}{3 \times 1.38 \times 10^{-23} \times 6.02 \times 10^{26}}
  • T = \frac{256 \times 10^6}{24.92 \times 10^3} \approx 10.27 \times 10^3 \text{ K}
  • T \approx 10^4 \text{ K}

\text{Result}

  • The required temperature is closest to K.
  • Correct Option: (a)

The Sigma Insight: Root Mean Square Speed

Solution Diagram

The Great Escape

Heating Hydrogen to the Limit
Imagine a tiny hydrogen molecule resting on the surface of the Earth. It's jiggling around, possessing a certain amount of kinetic energy due to its temperature. Now, what if we wanted to heat this molecule up so much that its random thermal motion becomes violent enough to break free from Earth's gravitational pull entirely?
This is a classic intersection of Thermodynamics and Gravitation. We are looking for the exact temperature where the root mean square (RMS) velocity of the gas molecule, , perfectly matches the escape velocity of the Earth, .

The Master Equations

From gravitation, we know that the escape velocity from the surface of a planet is given by:
where is the acceleration due to gravity and is the radius of the Earth.
On the other hand, the kinetic theory of gases tells us that the RMS velocity of a gas molecule at an absolute temperature is:
where is the Boltzmann constant and is the mass of a single molecule.

The Setup and The Trap

Our condition is simple: .
Equating the two expressions and squaring both sides to eliminate the square roots, we get:
Rearranging this to solve for our target variable, the temperature :
Now, here is where the examiners set a brilliant trap. We need the mass of a single hydrogen molecule, . Usually, we find this by dividing the molar mass by Avogadro's number ().
However, look closely at the units provided in the question: . This is not the standard Avogadro's number per mole; it is the number of molecules in a kilomole!
Since the molar mass of is , the mass of a single molecule is simply:

The Final Calculation

Substituting this expression for back into our temperature equation:
Now, we carefully plug in the given numerical values:
This value is closest to .
At this blistering temperature, the average hydrogen molecule is zipping around fast enough to escape Earth's gravity forever. This is precisely why our atmosphere retains heavier gases like nitrogen and oxygen, but lighter gases like hydrogen have long since boiled away into space!