Question 713: Thermodynamics & KTG

Question

If r.m.s. velocity of hydrogen molecules is 4 times that of an oxygen  molecule at $47^{\circ} \mathrm{C}$, the temperature of hydrogen molecules is (Molecular weight of Hydrogen and Oxygen are 2 and 32 respectively)

Accepted Answer

Given:
r.m.s. velocity of hydrogen $= 4 	imes$ r.m.s. velocity of oxygen
Temperature of oxygen, $T_{	ext{O}_2} = 47^\circ	ext{C} = 320\ 	ext{K}$
Molecular mass of Hydrogen, $M_{	ext{H}_2} = 2$
Molecular mass of Oxygen, $M_{	ext{O}_2} = 32$
Step 1: Formula for r.m.s. velocity
The r.m.s. velocity of a gas molecule is:
$$ v_{	ext{rms}} = \sqrt{\frac{3RT}{M}} $$
where
$R = $ universal gas constant
$T = $ absolute temperature (in K)
$M = $ molar mass (in kg/mol or g/mol as long as units are consistent)
Step 2: Ratio of r.m.s. velocities
Let temperature of hydrogen be $T_{	ext{H}_2}$.
Given:
$$ v_{	ext{rms, H}_2} = 4\, v_{	ext{rms, O}_2} $$
From the formula,
$$ \frac{v_{	ext{rms, H}_2}}{v_{	ext{rms, O}_2}} = \frac{\sqrt{\frac{3RT_{	ext{H}_2}}{M_{	ext{H}_2}}}} {\sqrt{\frac{3RT_{	ext{O}_2}}{M_{	ext{O}_2}}}} $$
This simplifies to,
$$ \frac{v_{	ext{rms, H}_2}}{v_{	ext{rms, O}_2}} = \sqrt{\frac{T_{	ext{H}_2} M_{	ext{O}_2}}{T_{	ext{O}_2} M_{	ext{H}_2}}} $$
Step 3: Substitute given values and solve for $T_{	ext{H}_2}$
Given, $\frac{v_{	ext{rms, H}_2}}{v_{	ext{rms, O}_2}} = 4$
So,
$$ 4 = \sqrt{ \frac{T_{	ext{H}_2} 	imes 32}{320 	imes 2} } $$
Now, square both sides:
$$ 16 = \frac{T_{	ext{H}_2} 	imes 32}{320 	imes 2} $$
$$ 16 = \frac{32 T_{	ext{H}_2}}{640} $$
$$ 16 = \frac{T_{	ext{H}_2}}{20} $$
So,
$$ T_{	ext{H}_2} = 16 	imes 20 = 320~	ext{K} $$
Step 4: Convert to degree Celsius
$$ T(^\circ \mathrm{C}) = T(\mathrm{K}) - 273 $$
$$ T_{	ext{H}_2} = 320 - 273 = 47^{\circ} \mathrm{C} $$
Final Answer:
$\boxed{47^{\circ} \mathrm{C}}$ (Option B)

Answer

$23^{\circ} \mathrm{C}$

Answer

$47^{\circ} \mathrm{C}$

Answer

$80^{\circ} \mathrm{C}$

Answer

$114^{\circ} \mathrm{C}$

Thermodynamics & KTG

If r.m.s. velocity of hydrogen molecules is 4 times that of an oxygen molecule at 47^{\circ} \mathrm{C}, the temperature of hydrogen molecules is (Molecular weight of Hydrogen and Oxygen are 2 and 32 respectively)

Choose the correct answer: