Question

When the temperature of an ideal gas is increased by 600 K, the velocity of the sound in the gas becomes $\sqrt 3$ times the initial velocity in it . The initial temperature of the gas is ${\text{in}}{{\text{ }}^\circ}C$
A. ${\text{- 7}}{{\text{3}}^\circ}C$
B. ${\text{2}}{{\text{7}}^\circ}C$
C. ${\text{12}}{{\text{7}}^\circ}C$
D. $327^\circ C$

Answer 1

The temperature of an ideal gas is directly proportional to the velocity of sound as more molecules will be colliding with each other with an increase in pressure too

Complete answer:
Let T be the initial temperature of an ideal gas and v is the initial velocity of sound. The speed of sound is proportional to the square root of the temperature, so according to the given situation of the problem
${\text{v}}\propto \sqrt {\text{T}} {\text{ and }}\sqrt {3v} \propto \sqrt {{\text{T + 600}}}$
$\Rightarrow \dfrac{{\sqrt 3 v}}{v}{\text{ = }}\sqrt {\dfrac{{{\text{T + 600}}}}{{\text{T}}}}$
$\Rightarrow \sqrt {\text{3}} {\text{ = }}\sqrt {\dfrac{{{\text{T + 600}}}}{{\text{T}}}}$
$\Rightarrow {\text{3 = }}\dfrac{{{\text{T + 600}}}}{{\text{T}}}$
$\Rightarrow 3{\text{T = T + 600}}$
$\therefore {\text{ T = 300 K}}$
Also We know .$0^\circ C = 273 K$
Hence T = 300 - 273
$\therefore T = 27^\circ C$

Hence, the answer to the above question is option B.

Additional Information:
Gases are complex. They are filled with billions and billions of energetic gas molecules that can collide and possibly interact with each other. Since an actual gas is difficult to describe accurately, people created the concept of an ideal gas as an approximation, which helps us model and predict the behavior of real gases. The term ideal gas refers to a hypothetical gas composed of molecules which follows few rule
Ideal gas molecules do not attract or repel each other. The only interaction between ideal gas molecules would be an elastic collision with each other or an elastic collision with the container walls.
Ideal gas molecules themselves do not take any quantity. The amount of gas increases as the molecule expands over a large area of ​​space, but the molecules of the ideal gas are produced as point particles that have no volume of their own.

Note: Ideal gases basically consist of particles which are at constant, random speeds. Their size is not usually considered as the particles are very small irrespective of the space between them. There are no gases that are absolutely ideal, but there are many gases that are close enough that the concept of an ideal gas is an extremely useful approximation for many situations.