The average kinetic energy of translation of a molecule of an ideal gas at temperature T is:a)(1/2) kTb)(5/7) kTc)(3/2) kTd)(7/2) kTCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Answer

Every degree of freedom has energy equal to 1/2KT

Ideal gas is monoatomic hence it has 3 degree of freedom

Therefore K.E.=3*1/2KT=3/2KT

Answer

The average kinetic energy of translation of a molecule of an ideal gas at temperature T can be calculated using the kinetic theory of gases. According to this theory, the average kinetic energy of a gas molecule is directly proportional to its temperature.

The kinetic energy of a molecule can be given by the equation:

KE = (1/2) mv^2

Where:
- KE is the kinetic energy
- m is the mass of the molecule
- v is the velocity of the molecule

Now, let's consider the average kinetic energy of translation of a molecule of an ideal gas at temperature T.

1. Deriving the equation
We know that the average kinetic energy of a gas molecule is directly proportional to its temperature. Mathematically, we can express this relationship as:

KE ∝ T

To find the constant of proportionality, we can use the ideal gas equation:

PV = nRT

Where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of the gas
- R is the ideal gas constant

We can rearrange the ideal gas equation to solve for the kinetic energy:

KE = (3/2) nRT

2. Relating the equations
Now, we need to relate the kinetic energy to the temperature in terms of the kinetic energy of translation. Since the kinetic energy is directly proportional to the temperature, we can write:

KE = kT

Where:
- k is a constant of proportionality

Comparing this equation with the previous equation derived from the ideal gas equation, we can conclude that:

k = (3/2) nR

Therefore, the average kinetic energy of translation of a molecule of an ideal gas at temperature T is:

KE = (3/2) kT

3. Simplifying the equation
To simplify the equation, we can substitute the value of the ideal gas constant R:

R = k/N_A

Where:
- N_A is Avogadro's number

Substituting this value into the equation, we get:

KE = (3/2) (k/N_A) T

Simplifying further, we have:

KE = (3/2) kT/N_A

Since N_A is a constant, we can rewrite the equation as:

KE = (3/2) kT

Therefore, the correct answer is option 'C': (3/2) kT.

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