At a temperature of 314 ,K and a pressure of | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given: Temperature $T = 314 \,K$, pressure $p = 100 \,kPa = 1.0 	imes 10^5 \,Pa$, speed of sound $v = 1380 \,ms^{-1}$, and radius of gas molecule

Vedclass · Accepted Answer

$1000 \sqrt{2} \,MHz$

Vedclass · Answer

(B) Given: Temperature $T = 314 \,K$, pressure $p = 100 \,kPa = 1.0 	imes 10^5 \,Pa$, speed of sound $v = 1380 \,ms^{-1}$, and radius of gas molecule $r = 0.5 \,Å = 0.5 	imes 10^{-10} \,m$. The diameter $d = 2r = 10^{-10} \,m$.The mean free path $\lambda$ is given by $\lambda = \frac{kT}{\sqrt{2} \pi d^2 p}$.The frequency $
u$ of the sound wave is given by $
u = \frac{v}{\lambda}$.Substituting $\lambda$ into the frequency formula, we get $
u = \frac{v \sqrt{2} \pi d^2 p}{kT}$.Substituting the given values:$
u = \frac{1380 	imes \sqrt{2} 	imes 3.14 	imes (10^{-10})^2 	imes 1.0 	imes 10^5}{1.38 	imes 10^{-23} 	imes 314}$$
u = \frac{1380 	imes \sqrt{2} 	imes 3.14 	imes 10^{-20} 	imes 10^5}{1.38 	imes 10^{-23} 	imes 314}$$
u = \frac{1380 	imes \sqrt{2} 	imes 3.14 	imes 10^{-15}}{1.38 	imes 314 	imes 10^{-23}}$$
u = \frac{1380 	imes \sqrt{2} 	imes 3.14 	imes 10^8}{433.32} \approx 10 	imes \sqrt{2} 	imes 10^8 \,Hz = \sqrt{2} 	imes 10^9 \,Hz = 1000 \sqrt{2} \,MHz$.Thus, the correct option is $B$.

Similar Questions

The mean free path of molecules of a gas (radius $r$) is inversely proportional to

Calculate the value of mean free path $(\lambda)$ for oxygen molecules at temperature $27^{\circ} C$ and pressure $1.01 \times 10^{5} Pa$. Assume the molecular diameter $d = 0.3 nm$ and the gas is ideal. Given Boltzmann constant $k = 1.38 \times 10^{-23} J K^{-1}$. (Result in $nm$)

If the radius of an $O_{2}$ molecule is $40 \; \mathring{A}$,$T = 27^{\circ}C$,and $P = 1 \; \text{atm}$,find the relaxation time.

How does the mean free path of a gas depend on temperature?

Consider a gas for which the diameter of molecules is $\sigma$. The gas is at a pressure $P$ and temperature $T$,and $N_a$ is Avogadro's number. What is the mean free path along the $x$-axis?

Vedclass Test Series

Exam Paper Generator

Online Exam Module