Closed at one end and fitted with a piston at the other, an adiabatic cylindrical tube has a cross-sectional area of 1 cm ^2 . The tube holds 0.03 g of an ideal gas. At a pressure of 1 atm and at the surrounding temperature, the gas column has a length of 40 cm . Suddenly, the piston is pulled out, doubling the length of the column, which causes the gas pressure to drop to 0.355 atm . Determine the speed of sound in

Step-by-step solution

For an adiabatic process, the relation between pressure and volume is given by P₁ V₁^ = P₂ V₂^ . Given P₁ = 1 atm , V₁ = A L₁ = 1 40 = 40 cm ^3 . After the piston is pulled out, P₂ = 0.355 atm and V₂ = A L₂ = 1 80 = 80 cm ^3 . Substituting the values: 1 (40)^ = 0.355 (80)^ 2^ = 1 0.355 2.817 Since 2^ 1.5 = 2 2 2.828 , we get 1.5 . The density of the gas at atmospheric temperature is: = m V₁ = 0.03 10⁻³ 40 10⁻⁶ = 0.75 kg/m ^3 The speed of sound in the gas is given by: v = P₁ Taking 1 atm = 10^5 Pa : v = 1.5 10^5 0.7