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24 23 2. (25 pts.) A sample of helium gas contains 3.0115 x 1024 atoms. (NA = 6.023 x 107) (a) Determine the mass of the sample in grams. (b) Find its volume at STP--standard temperature (0° C.) & standard pressure (1 atm). (c) If the pressure increases to a value of 5 atm, while the temperature is decreased to 182° C., then determine the new volume of this same sample. (d) Suppose now that the container for this sample of helium gas develops a leak overnight while it is fixed at a constant volume. If the gauge pressure of the container drops from 9 to 4 atm during this period while the temperature drops from 27 to insulation to be 3° C., then determine the percentage of the original gas still remaining. 3. (25 pts.) (a) A sample of 3 moles of an ideal gas expands isothermally such that its volume doubles. What happens to its pressure and its internal energy? (b) If the same gas sample expands adiabatically instead, and does 40 J of work during the expansion, determine the change in internal energy (include sign also). (c) If the same gas sample expands isobarically instead, at 1 atm (1.013 x 105 Pa), then determine the net work done by the gas if the volume changes from 10 L to 20 L. (d) If a heat engine operates in a cycle between the fixed input and output temperatures of 100°C and 0°C, determine the maximum possible efficiency. (e) If 500 J of work are to be done by this engine, determine the heat input to this system, assuming it is operating at the maximum efficiency found in part (d). 4. (25 pts.) A 4-gram string of length 3 m is fixed at both ends, and standing waves are generated in the string. The value of the tension in the string is 20 N. (a) If the whole string vibrates in 3 loops, find the wavelength of the waves in the string. Identify the harmonic and overtone number of this vibration and sketch it. Label clearly the positions of all nodes and antinodes in your sketch.