If the absolute temperature of an ideal gas increases from T₁ to T₂, then the change in average internal energy of the gas molecule is given as: (k_B = Boltzmann constant)

CONCEPT:

Kinetic interpretation of temperature:

• We know that pressure P of an ideal gas is given as,

• If the volume of the gas is V, then,

∵ N = nV

• The internal energy of an ideal gas is purely kinetic.
• So the total internal energy E of an ideal gas is given as,

• So we can say,

Where n = number of molecules per unit volume, m = mass of the molecule, N = total number of molecules, and  = mean of the squared speed

• If the absolute temperature of an ideal gas is T, then the total internal energy is given as,

Where kB = Boltzmann constant

• So the average kinetic energy of a molecule is given as,

• The average kinetic energy of a molecule is proportional to the absolute temperature of the gas.
• The average kinetic energy of a molecule is independent of pressure, volume, or the nature of the ideal gas.
• So we can say that the internal energy of an ideal gas depends only on temperature, not on pressure or volume.

EXPLANATION:

• We know that if the absolute temperature of an ideal gas is T, then the average internal energy of a molecule is given as,

     -----(1)

Where kB = Boltzmann constant and T = absolute temperature

By equation 1 the average internal energy of a molecule for temperature T₁ is given as,

     -----(2)

By equation 1 the average internal energy of a molecule for temperature T2 is given as,

     -----(3)

By equation 2 and equation 3,

• Hence, option 1 is correct.

Additional Information

Dalton’s law of partial pressures:

• For a mixture of non-reactive ideal gases, the total pressure gets contribution from each gas in the mixture.

• In equilibrium, the average kinetic energy of the molecules of different gases will be equal.

So,

⇒ P = (n1 + n2 + ... + nn)kBT

• The mean of the square speed is given as,

• The square root of  is known as root mean square (RMS) speed and is denoted by vrms.
• At the same temperature, lighter molecules have a greater RMS speed.