The parameter that remains the same for molecules of all gases at a given temperature is :
- kinetic energy
- momentum
- mass
- speed
The Correct Option is A
Approach Solution - 1
To determine which parameter remains the same for molecules of all gases at a given temperature, one must understand the relationship between the temperature of a gas and the kinetic theory of gases.
The kinetic theory of gases describes a gas as a large number of small particles (atoms or molecules), all of which are in constant, random motion. This theory is based on several assumptions, one of which is that the temperature of a gas is directly proportional to the average kinetic energy of its molecules.
- Kinetic Energy: According to the kinetic theory of gases, the average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas. Mathematically, this is expressed as: \(E_k = \frac{3}{2}kT\) where \(E_k\) is the average kinetic energy, \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature. This implies that, at a given temperature, the average kinetic energy is the same for molecules of all gases.
- Momentum: The momentum of a molecule is given by the product of its mass and velocity \((p = mv)\). Since the masses and velocities of gas molecules vary, momentum does not remain constant across different gases at a given temperature.
- Mass: The mass of a molecule is determined by the type of gas. Different gases have different molecular masses. Therefore, mass is not a parameter that remains the same for molecules of all gases.
- Speed: The speed of the gas molecules depends on their kinetic energy and mass. Since different gases have different masses, the speed of the molecules will vary even if their kinetic energy is the same.
In conclusion, among the options given, the parameter that remains the same for molecules of all gases at a given temperature is the kinetic energy. This is consistent with the principle that temperature is a measure of the average kinetic energy of the molecules in a gas.
Approach Solution -2
The kinetic energy of gas molecules at a given temperature is given by:
\[ KE = \frac{f}{2}kT \]
where \( f \) is the degrees of freedom and \( kT \) is the thermal energy term.
This is a conceptual fact based on the equipartition theorem, which states that all gases at a given temperature have the same average kinetic energy per molecule.
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