Question 13: The temperature of a gas is -50^{\circ}\mathrm{C}. To what temperature the gas should be heated so that the rms speed is increased by 3 times?
(1) 3097 K
(2) 223 K
(3) 669^{\circ}\mathrm{C}
(4) 3295^{\circ}\mathrm{C}
Answer: Option (4)
Explanation:
The rms speed of gas molecules is directly proportional to the square root of absolute temperature.
v_{\mathrm{rms}} \propto \sqrt{T}The initial temperature of the gas is
T_{1}=-50^{\circ}\mathrm{C}+273=223\,\mathrm{K}The rms speed is increased by 3 times, which means the final rms speed becomes
v_{2}=v_{1}+3v_{1}=4v_{1}Using the relation between rms speed and temperature,
\frac{v_{2}}{v_{1}}=\sqrt{\frac{T_{2}}{T_{1}}}Substituting values,
4=\sqrt{\frac{T_{2}}{223}}Squaring both sides,
16=\frac{T_{2}}{223} T_{2}=16\times223=3568\,\mathrm{K}Converting this temperature into degree Celsius,
T_{2}=3568-273=3295^{\circ}\mathrm{C}Hence, the correct answer is Option (4).