velocity of sound in air in 27C is 340m/s what will be the velocity if the temperature is increased by 50C and pressure of air is doubled?

T

_{1}= 300 K

T

_{2}= 350 K

v

_{1}= 340 ms

^{-1}

P

_{1}= P

P

_{2 }= 2P

Let initial density = ρ

_{1}

Final density = ρ

_{2}

Assuming a perfect gas law remains valid in the atmosphere,

The equation of state of the gas is given by

M = equivalent molar mass of air

P

_{1}V

_{1}= nRT

_{1}

=> P

_{1}V

_{1 }= (m/M) RT

_{1}

=> P

_{1}= (m/V

_{1})RT

_{1}/M

=> P

_{1}= ρ

_{1}RT

_{1}/M

=> P

_{1}/ρ

_{1}= RT

_{1}/M

For the first state

P/ρ

_{1}= RT

_{1}/M

Multiplying both sides by γ

γP/ρ

_{1}= γRT

_{1}/M

v

_{1}= √ (γP/ρ

_{1})

=> v

_{1}= √ (γRT

_{1}/M) ---1.

For the second state

2P/ρ

_{2}= RT

_{2}/M

=> ρ

_{2}= 2PM/(RT

_{2})

Now,

v

_{2}= √(γP

_{2}/ρ

_{2})

=> v

_{2}= √(γ2P/ρ

_{2})

=> v

_{2}= √(γ2P/{2PM/(RT

_{2})})

=> v

_{2}= √(γRT

_{2}/M) ---2.

v

_{2}/v

_{1}= √(T

_{2}/T

_{1})

v

_{2}= v

_{1}√(T

_{2}/T

_{1})

= 340×√(350/300)

≈ 367 m/s

**
**