EXAMPLE 10.10 A Cylinder of Helium GOAL Calculate the internal energy of a system and the average kinetic energy per molecule. PROBLEM A cylinder contains 2.00 mol of helium gas at 20.0C. Assume the helium behaves like an ideal gas. (a) Find the total internal energy of the system. (b) What is the average kinetic energy per molecule? (c) How much energy would have to be added to the system to double the rms speed? The molar mass of helium is equal to 4.00 x 10'3 kg/mol. STRATEGY This problem requires substitution of given information into the appropriate equations: the internal energy of monatomic gas equation for part (a) and average kinetic energy equation for part (b). In part (c) use the equations for the rms speed and internal energy together. A change in the internal energy must be computed. SOLUTION (A) Find the total internal energy of the system. Substitute values into the internal U = 3(2'00 \"10003.31 J/mol _ K)(293 K) = 730 x 103 energy of monatomic gas equation with 2 n = 2.00 and T = 293 K. J (B) What is the average kinetic energy per molecule? Substitute given values into the impz = ikBT = 1(138 x 10-23 J/K)(293 K) = average kinetic energy equation. 2 2 2 6.07 x 10'21 J (C) How much energy must be added to double the rms speed? From the root mean square speed AU = U _ U. = inRT _ inRT. = inR(T _ T.) equation, doubling the rms speed f I 2 f 2 i 2 f 1 requires quadrupling T. Calculate the required change of internal energy, which is the energy that must be put 293 K] into the system. AU = 9.00 mo|)(8.31 J/mol - K)[(4.UD x 293 K) = 2.19 x 104 J QUESTION At the same temperature, how does the internal energy of 1 mole of helium gas compare with the internal energy of 1 mole of argon gas? (Select all that apply.) C] Although the helium atoms have a smaller mass m, they move faster, and have more internal energy because 1/2 mv2 depends linearly on the mass but quadratically on the speed. C] They are the same, partly because both contain the same number of atoms. C] The argon has more internal energy, partly because its atomic mass and therefore its average kinetic energy is greater. C] The helium has more internal energy, partly because the average speed of its atoms is greater. C] They are the same, partly because the average speed of the argon atoms and of the helium atoms are equal. C] They are the same, partly because both have the same average kinetic energy per atom. PRACTICE IT Use the worked example above to help you solve this problem. A cylinder contains 1.95 mol of helium gas at 175C. Assume that the helium behaves like an ideal gas. (a) Find the total internal energy of the system. J H (b) What is the average kinetic energy per molecule? J U (c) How much energy would have to be added to the system to double the rms speed? The molar mass of helium is 4.00 x 10'3 kg/mol. EXERCISE HINTS: GETTINGSTARTED | I'M STUCK! The temperature of 6.00 moles of argon gas is lowered from 2.60 x 102 K to 2.20 x 102 K. (a) Find the change in the internal energy, AU, of the gas. J H (b) Find the change in the average kinetic energy per atom. :11