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An observed gas's temperature is a function of the molecules' average translational kinetic energy.

Here $v_{\mathrm{rms}}鈭�\sqrt{T}$, then $T鈭�v_{\mathrm{rms}}^2$so doubling the speed of each molecule (which would double $v_{rms}$) quadruples the temperature, which implies that the temperature increases by the factor of $2^2.$

Therefore, the temperature increases by the factor of$2^2.$

Molar specific heat at constant volume is the amount of heat needed to raise the temperature of $1mole$of a gas through $1K$at constant volume, $CV.$

Temperature increases when molecules double in speed.

It is not dependent on the speed of individual molecules but rather the characteristics of those molecules, e.g. whether they are monoatomic, diatomic etc.

The molar specific heat at constant volume is independent of all of this.

When the volume remains the same, the molar specific heat will remain the same.