This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
To find the speed of sound in a gas mixture, we use the formula: v = sqrt(_mix) R TM_mix where _mix is the effective adiabatic index of the mixture, R is the universal gas constant, T is the absolute temperature, and M_mix is the effective molar mass of the mixture. Step 1: Convert the temperature to Kelvin. Given temperature T = 27^. T = 27 + 273.15 = 300.15 K ≈ 300 K Step 2: Calculate the molar masses of Hydrogen (H_2) and Oxygen (O_2). Molar mass of H = 1 g/mol. Molar mass of O = 16 g/mol. M_H_2 = 2 × 1 = 2 g/mol = 0.002 kg/mol M_O_2 = 2 × 16 = 32 g/mol = 0.032 kg/mol Step 3: Calculate the effective molar mass (M_mix) of the mixture. Number of moles of H_2 (n_H_2) = 4 moles Number of moles of O_2 (n_O_2) = 1 mole Total moles (n_total) = 4 + 1 = 5 moles The effective molar mass is given by: M_mix = n_H_2 M_H_2 + n_O_2 M_O_2n_total M_mix = (4 mol × 0.002 kg/mol) + (1 mol × 0.032 kg/mol)5 mol M_mix = 0.008 kg + 0.032 kg5 mol M_mix = 0.040 kg5 mol = 0.008 kg/mol Step 4: Determine the effective adiabatic index (_mix) for the mixture. Both Hydrogen (H_2) and Oxygen (O_2) are diatomic gases. For diatomic gases, the adiabatic index = (C_p)/(C_v) = (7/2 R)/(5/2 R) = (7)/(5) = 1.4. Since both components of the mixture are diatomic, the effective adiabatic index for the mixture will also be 1.4. _mix = 1.4 Step 5: Calculate the speed of sound. Given R = 8.31 J mol^-1 K^-1. v = sqrt(_mix) R TM_mix v = sqrt(1.4 × 8.31 J mol)^-1 K^-1 × 300 K0.008 kg/mol v = sqrt((3490.2)/(0.008)) m/s v = sqrt(436275) m/s v ≈ 660.51 m/s Comparing this value with the given options: (A) 600 ms^-1 (B) 660 ms^-1 (C) 630 ms^-1 (D) 700 ms^-1 The calculated speed of sound is approximately 660 ms^-1. The final answer is (B). Last free one today — make it count tomorrow, or type /upgrade for unlimited.