Temperature Dependence Of Speed

Since the root-mean-square speed scales as $v_{rms}\propto \sqrt{T}$ at fixed molar mass, doubling the speed requires the absolute temperature to increase by a factor of four. This square-root dependence arises because kinetic energy is proportional both to $v^2$ and to $T$, so the speed itself grows only as the square root of temperature. Writing $\frac{v_2}{v_1}=\sqrt{\frac{T_2}{T_1}}$ and setting $v_2=2v_1$ gives $\frac{T_2}{T_1}=4$. With $T_1=300$ K, the required temperature is $T_2=4\times 300=1200$ K. It is essential to use absolute kelvin temperature here, since the proportionality fails on the Celsius scale. The value 600 K wrongly assumes a linear relationship between speed and temperature. The value 900 K corresponds to tripling the temperature, which would raise the speed by only $\sqrt{3}\approx 1.73$. The value 2400 K applies the scaling factor twice. This reflects the NCERT relation between molecular speed and absolute temperature, where speed grows with the square root of temperature, not linearly, so very high temperatures are needed to achieve modest gains in molecular speed. As a plausibility check, quadrupling the temperature quadruples the average kinetic energy and therefore doubles the speed, fully consistent with the result of 1200 K.