Question

The RMS velocity of Co, at a temperature
"T" is x cm/sec.At what tempeature, the RMS
velocity of CO, would be 4x
1) 161 2) 2T 3) 4T 4) 32T​

Answer 1

GIVEN :–

• RMS velocity of Co, at a temperature "T" is x cm/sec.

TO FIND :–

• At what tempeature, the RMS velocity of Co, would be 4x ?

SOLUTION :–

• We know that –

ㅤ

$\bf \implies C_{R.M.S.} = \sqrt{ \dfrac{3RT}{M}}$

ㅤ

$\bf \implies { \boxed{ \bf C_{R.M.S.} = \sqrt{ \dfrac{3RT}{M}}}}$

ㅤ

• According to first condition –

ㅤ

$\bf \implies C_{R.M.S.} = \sqrt{ \dfrac{3RT}{28}}$

ㅤ

$\bf \implies x = \sqrt{ \dfrac{3RT}{28}} \:\:\:-----eq(1)$

ㅤ

• According to the second condition –

ㅤ

$\bf \implies C_{R.M.S.} = \sqrt{ \dfrac{3RT'}{28}}$

ㅤ

$\bf \implies 4x = \sqrt{ \dfrac{3RT'}{28}} \:\:\:-----eq(2)$

ㅤ

• Now eq.(1) ÷ eq.(2) –

ㅤ

$\bf \implies \dfrac{x}{4x} = \dfrac{ \sqrt{ \dfrac{3RT}{28}}}{\sqrt{ \dfrac{3RT'}{28}}}$

ㅤ

$\bf \implies \dfrac{1}{4} = \dfrac{ \sqrt{ \dfrac{T}{28}}}{\sqrt{ \dfrac{T'}{28}}}$

ㅤ

• Square on both sides –

ㅤ

$\bf \implies \dfrac{1}{16} = \dfrac{\left(\dfrac{T}{28}\right)}{\left(\dfrac{T'}{28}\right)}$

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$\bf \implies \dfrac{1}{16} \times \dfrac{T'}{28}= \dfrac{T}{28}$

ㅤ

$\bf \implies T'= \dfrac{28T}{28} \times 16$

ㅤ

$\bf \implies T'= 16T$

ㅤ

$\bf \implies \large{ \boxed{ \bf T' = 16T}}$

▪︎Hence , Option (1) is correct.

Answer 2

$\huge\red{\bold{\underline{\underline{Given:-}}}}$

• RMS velocity of CO at certain temperature "T" is 'x' cm/sec

$\huge\blue{\bold{\underline{\underline{To\:Find:-}}}}$

• Temperature at which RMS velocity is 4x

$\huge\pink{\bold{\underline{\underline{Solution:-}}}}$

Ratio of RMS velocities of same gas at two different temperatures is given by ,

$\large\rm\bold{\boxed{{u}_</p><p>{rms}\:=\:\sqrt{\frac{3RT}{M}}}}$

Where ,

• R is universal gas constant
• M is molecular mass
• T is temperature

CASE - I :-

R = 8.314 × 10⁷ erg.mol⁻¹.K⁻¹

T = T

u₁ = x cm/sec

Mass of Co = 59g/mol

CASE - II :-

R = 8.314 erg.mol⁻¹.K⁻¹

T = T₂ ( let )

u₂ = 4x cm/sec

Mass of CO = 28.01 g/mol

Ratio of RMS velocities of Co and CO is

$\large\rm{\rightarrow{\frac{x}{4x}\:=\:\sqrt{\frac{\frac{T}{59}}{\frac{T_2}{28.01}}}}}$

$\large\rm{\rightarrow{\frac{1}{4}\:=\:\sqrt{\frac{T\times\:28.01}{T_2\times\:59}}}}$

Squaring on both sides ,

$\large\rm{\rightarrow{\frac{1}{16}\:=\:{\frac{T\times\:28.01}{T_2\times\:59}}}}$

$\large\rm{\rightarrow{\frac{1}{16}\:=\:\frac{T}{T_2}\times\:2}}$

$\large\rm{\rightarrow{32\:\approx\:T_2}}$

∴ 32T is the temperature at which the RMS velocity of CO is 4x

$\huge\color{teal}{\bold{\underline{\underline{Additional\:Info:-}}}}$

➣ Ratio of RMS velocities of different gases at same temperature is given by ,

$\large\rm\bold{\boxed{\frac{u_1}{u_2}\:=\:\sqrt{\frac{M_2}{M_1}}}}$

➣ RMS velocity when temperature and mass of gas are given , is given by

$\large\rm\bold{\boxed{{u}_{rms}\:=\:\sqrt{\frac{3RT}{M}}}}$