Temperature Dependence Of Speed

Question

A certain gas sample has a root-mean-square molecular speed v when its temperature is 300 K; at what temperature will this speed become 2v?

RankGuru · Accepted Answer

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 $\dfrac{v_2}{v_1}=\sqrt{\dfrac{T_2}{T_1}}$ and setting $v_2=2v_1$ gives $\dfrac{T_2}{T_1}=4$. With $T_1=300$ K, the required temperature is $T_2=4	imes300=1200$ 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 $\sqrt{3}\approx1.73$. 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.

Temperature Dependence Of Speed

Select the correct option:

🔒 Solution Hidden from View

More kinetic theory of gases Practice Questions

How does the mean free path of molecules in a gas sample change if the number density of molecules i...

In a laboratory vacuum chamber the pressure of a gas is reduced to one-tenth of its initial value wh...

A weather balloon contains 2 moles of an ideal gas, so approximately how many gas molecules are pres...

Equal volumes of two different ideal gases are compared at the same temperature and pressure, so wha...

As the temperature of a gas sample is increased, how does the shape of its Maxwell molecular speed d...

A rigid insulated tank contains a mixture of 2 moles of a monatomic gas and 2 moles of a diatomic ga...

Which of the following assumptions is fundamental to the kinetic theory model of an ideal gas as des...

Two containers hold helium and oxygen respectively at the same absolute temperature, so how do the a...

When the absolute temperature of a fixed sample of ideal gas is raised from 300 K to 600 K, by what ...

Consider a tank of nitrogen molecules each of molar mass 28 grams per mole maintained at 300 K, so w...

Hydrogen and oxygen samples are kept at the same temperature, and given that oxygen is sixteen times...

For a sample of ideal gas at a given temperature, arrange the most probable speed, the average speed...

About This Question

Solution