Question

The temperature of a sample of an ideal gas becomes double. Which of the following statements best describes the result of this temperature change?

• A. The root-mean-square speed of the gas molecules doubles
• B. The average kinetic energy of the gas molecules increases by a factor of $\sqrt{2}$
• C. The average kinetic energy of the gas molecules increases by a factor of $4$
• D. The speeds of the gas molecules cover a wide range, but the root-mean-square speed increases by a factor of $\sqrt{2}$
• E. The speeds of the gas molecules cover a wide range, but the root-mean-square speed increases by a factor of $2$

Answer 1

Answer: (D)

The speeds of the gas molecules cover a wide range, but the root-mean-square speed increases by a factor of $\sqrt{2}$

Root mean square speed is a statistical measurement of speed. It is not the actual speed of each gas molecule, but a mathematical estimate of root of sum of squares of the speeds of molecules, which is given by $\sqrt{\frac{3RT}{M_0}}$.

Thus when temperature becomes double, the root mean square speed increases by a factor of $\sqrt{2}$.