15.4 Use the formula $\mathit{v}=\sqrt{\frac{\mathit{\gamma }\mathit{P}}{\mathit{\rho }}}\mathit{}\mathit{}$ to explain why the speed of sound in air

(a) Is independent of pressure

(b) Increases with temperature

(c) Increases with humidity

<p>In the equation&nbsp;<span title="Click to copy mathml"><math><mi>v</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mrow><mi>γ</mi><mi>P</mi></mrow></mrow><mrow><mrow><mi>ρ</mi></mrow></mrow></mfrac></mrow></msqrt></math></span>&nbsp;……(i)</p><p><span title="Click to copy mathml"><math><mi>ρ</mi></math></span>&nbsp;Density =&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>M</mi><mi>a</mi><mi>s</mi><mi>s</mi></mrow></mrow><mrow><mrow><mi>V</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>m</mi><mi>e</mi></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>M</mi></mrow></mrow><mrow><mrow><mi>V</mi></mrow></mrow></mfrac></math></span>&nbsp;where M = molecular weight of the gas, V = Volume of the gas, so we can write</p><p><span title="Click to copy mathml"><math><mi>v</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mrow><mi>γ</mi><mi>P</mi><mi>V</mi></mrow></mrow><mrow><mrow><mi>M</mi></mrow></mrow></mfrac></mrow></msqrt></math></span>&nbsp;…….(ii)</p><p>For ideal gas equation, PV = nRT, n = 1 so PV = RT</p><p>For constant T, PV = constant</p><p>In equation (ii), since PV = constant,&nbsp;<span title="Click to copy mathml"><math><mi>γ</mi></math></span>&nbsp;and M constant, v is also constant. Hence, at a constant temperature, the speed of sound in a gaseous medium is independent of the change in the pressure of the gas.</p><p>From equation (i)&nbsp;<span title="Click to copy mathml"><math><mi>v</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mrow><mi>γ</mi><mi>P</mi></mrow></mrow><mrow><mrow><mi>ρ</mi></mrow></mrow></mfrac></mrow></msqrt></math></span></p><p>For 1 mole of an ideal gas, the gas equation can be written as PV = RT or P =&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>R</mi><mi>T</mi></mrow></mrow><mrow><mrow><mi>V</mi></mrow></mrow></mfrac></math></span></p><p>Substituting in equation (i), we get&nbsp;<span title="Click to copy mathml"><math><mi>v</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mrow><mi>γ</mi><mi>R</mi><mi>T</mi></mrow></mrow><mrow><mrow><mi>ρ</mi><mi>V</mi></mrow></mrow></mfrac></mrow></msqrt></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msqrt><mrow><mfrac><mrow><mrow><mi>γ</mi><mi>R</mi><mi>T</mi></mrow></mrow><mrow><mrow><mi>M</mi></mrow></mrow></mfrac></mrow></msqrt></math></span></p><p>Since&nbsp;<span title="Click to copy mathml"><math><mi>γ</mi></math></span>&nbsp;, R and M are constant, we get v&nbsp;<span title="Click to copy mathml"><math><mo>∝</mo><mo>√</mo><mi>T</mi></math></span></p><p>Hence, the speed of sound in a gas is directly proportional to the square root of the temperature of the gaseous medium. i.e. the speed of sound increase with an increase in the temperature of the gaseous medium and vice versa.</p><p>Let&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>m</mi></mrow></mrow></msub></math></span>&nbsp;and&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>d</mi></mrow></mrow></msub></math></span>&nbsp;be the speed of sound in moist air and dry air respectively and&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>m</mi><mi></mi></mrow></mrow></msub></math></span>&nbsp;and&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>d</mi></mrow></mrow></msub></math></span>&nbsp;be the corresponding densities</p><p>From equation (i)&nbsp;<span title="Click to copy mathml"><math><mi>v</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mrow><mi>γ</mi><mi>P</mi></mrow></mrow><mrow><mrow><mi>ρ</mi></mrow></mrow></mfrac></mrow></msqrt></math></span>&nbsp;, we get&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi></mi><mi>v</mi></mrow></mrow><mrow><mrow><mi>m</mi></mrow></mrow></msub><mi mathvariant="normal"></mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mrow><mi>γ</mi><mi>P</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>m</mi><mi></mi></mrow></mrow></msub></mrow></mrow></mfrac></mrow></msqrt></math></span>&nbsp;and&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi></mi><mi>v</mi></mrow></mrow><mrow><mrow><mi>d</mi></mrow></mrow></msub><mi mathvariant="normal"></mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mrow><mi>γ</mi><mi>P</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>d</mi><mi></mi></mrow></mrow></msub></mrow></mrow></mfrac></mrow></msqrt></math></span></p><p><span title="Click to copy mathml"><math><mfrac><mrow><mrow><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>m</mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>d</mi></mrow></mrow></msub></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msqrt><mrow><mfrac><mrow><mrow><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>d</mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>m</mi><mi></mi></mrow></mrow></msub></mrow></mrow></mfrac></mrow></msqrt></math></span></p><p>However, the presence of water vapour reduces the density of air, i.e.&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>d</mi></mrow></mrow></msub><mo>&lt;</mo></math></span>&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>m</mi><mi></mi></mrow></mrow></msub></math></span>&nbsp;, so&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>m</mi></mrow></mrow></msub><mo>&lt;</mo></math></span>&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>d</mi></mrow></mrow></msub></math></span></p>

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