Question

The temperature of a gas is $-68^\circ$C. To what temperature should it be heated, so that the r.m.s. velocity of the molecules becomes doubled?

• A. $357^\circ$C
• B. $457^\circ$C
• C. $547^\circ$C
• D. $820^\circ$C

Hint:

Physics Tip : Gas law temperature formulas always use Kelvin scale.

Answer 1

The Correct Option is C

Step 1: Relation of r.m.s. speed with temperature. For an ideal gas, root mean square speed is: $$ v_{\text{rms}} \propto \sqrt{T} $$ where $T$ is absolute temperature in Kelvin.

Step 2: Convert initial temperature into Kelvin. Given: $$ T_1=-68^\circ C $$ So, $$ T_1=273-68=205\text{ K} $$

Step 3: Speed is doubled. If final speed becomes double: $$ v_2=2v_1 $$ Using proportionality: $$ \frac{v_2}{v_1}=\sqrt{\frac{T_2}{T_1}} $$ $$ 2=\sqrt{\frac{T_2}{205}} $$

Step 4: Square both sides. $$ 4=\frac{T_2}{205} $$ $$ T_2=820\text{ K} $$

Step 5: Convert back to Celsius. $$ T_2=820-273=547^\circ C $$ $$ \therefore \text{Correct option is (C) }547^\circ C. $$