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.
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Here's how to solve the problem: Given: Molar mass of gas M = 0.024 \, kg/mol Pressure P = 1.40 × 10^5 \, N/m^2 Temperature T_C = 78 \, ^ C Cylinder height h = 7.2 \, cm = 0.072 \, m Cylinder diameter d = 2.5 \, cm = 0.025 \, m Constants: Ideal gas constant R = 8.314 \, J/(mol · K) Avogadro's number N_A = 6.022 × 10^23 \, mol^-1 Boltzmann constant k = 1.38 × 10^-23 \, J/K Step 1: Calculate the volume of the container and convert temperature to Kelvin. The radius of the cylinder is r = (d)/(2) = 0.025 \, m2 = 0.0125 \, m. The volume V of the cylinder is: V = r^2 h = (0.0125 \, m)^2 (0.072 \, m) V = (0.00015625 \, m^2) (0.072 \, m) = 0.0000353429 \, m^3 Convert the temperature from Celsius to Kelvin: T = T_C + 273.15 = 78 \, ^ C + 273.15 = 351.15 \, K Part (a) i) What mass of gas is present in the container? Step 2: Calculate the number of moles (n) of gas using the ideal gas law. The ideal gas law is PV = nRT. Rearranging for n: n = (PV)/(RT) n = (1.40 × 10^5 \, N/m^2) × (0.0000353429 \, m^3)(8.314 \, J/(mol · K)) × (351.15 \, K) n = (4.948006)/(2919.9861) = 0.0016945 \, mol Step 3: Calculate the mass (m) of the gas. The mass of the gas is the number of moles multiplied by the molar mass: m = n × M m = 0.0016945 \, mol × 0.024 \, kg/mol m = 0.000040668 \, kg Rounding to three significant figures: m = 4.07 × 10^-5 \, kg Part (a) ii) How many molecules of the gas are present in the container? Step 4: Calculate the number of molecules (N). The number of molecules is the number of moles multiplied by Avogadro's number: N = n × N_A N = 0.0016945 \, mol × 6.022 × 10^23 \, mol^-1 N = 1.0203 × 10^21 \, molecules Rounding to three significant figures: N = 1.02 × 10^21 \, molecules Part (b) Calculate the average kinetic energy of the gas molecules. Step 5: Calculate the average kinetic energy (KE_avg) of the gas molecules. For an ideal gas, the average translational kinetic energy per molecule is given by: KE_avg = (3)/(2) kT KE_avg = (3)/(2) × (1.38 × 10^-23 \, J/K) × (351.15 \, K) KE_avg = 1.5 × 1.38 × 10^-23 × 351.15 \, J KE_avg = 7.2798 × 10^-21 \, J Rounding to three significant figures: KE_avg = 7.28 × 10^-21 \, J Drop the next question!