Chapter 19

a) What is the root-mean-square speed for a collection of helium- 4 atoms at \(300 . \mathrm{K} ?\) b) What is the root-mean-square speed for a collection of helium- 3 atoms at 300 . K?

Two isotopes of uranium, \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U},\) are separated by a gas diffusion process that involves combining them with flourine to make the compound \(\mathrm{UF}_{6} .\) Determine the ratio of the root-mean-square speeds of UF \(_{6}\) molecules for the two isotopes. The masses of \({ }^{235} \mathrm{UF}_{6}\) and \({ }^{238} \mathrm{UF}_{6}\) are \(249 \mathrm{amu}\) and \(252 \mathrm{amu}\).

The electrons in a metal that produce electric currents behave approximately as molecules of an ideal gas. The mass of an electron is \(m_{\mathrm{e}} \doteq 9.109 \cdot 10^{-31} \mathrm{~kg} .\) If the temperature of the metal is \(300.0 \mathrm{~K},\) what is the root-mean-square speed of the electrons?

In a period of \(6.00 \mathrm{~s}, 9.00 \cdot 10^{23}\) nitrogen molecules strike a section of a wall with an area of \(2.00 \mathrm{~cm}^{2}\). If the molecules move with a speed of \(400.0 \mathrm{~m} / \mathrm{s}\) and strike the wall head on in elastic collisions, what is the pressure exerted on the wall? (The mass of one \(\mathrm{N}_{2}\) molecule is \(4.68 \cdot 10^{-26} \mathrm{~kg}\).)

Assuming the pressure remains constant, at what temperature is the root-mean- square speed of a helium atom equal to the root-mean-square speed of an air molecule at STP?

At room temperature, identical gas cylinders contain 10 moles of nitrogen gas and argon gas, respectively. Determine the ratio of energies stored in the two systems. Assume ideal gas behavior.

Calculate the change in internal energy of 1.00 mole of a diatomic ideal gas that starts at room temperature \((293 \mathrm{~K})\) when its temperature is increased by \(2.00 \mathrm{~K}\).

Treating air as an ideal gas of diatomic molecules, calculate how much heat is required to raise the temperature of the air in an \(8.00 \mathrm{~m}\) by \(10.0 \mathrm{~m}\) by \(3.00 \mathrm{~m}\) room from \(20.0^{\circ} \mathrm{C}\) to \(22.0^{\circ} \mathrm{C}\) at \(101 \mathrm{kPa}\). Neglect the change in the number of moles of air in the room.

What is the approximate energy required to raise the temperature of \(1.00 \mathrm{~L}\) of air by \(100 .{ }^{\circ} \mathrm{C} ?\) The volume is held constant.

You are designing an experiment that requires a gas with \(\gamma=1.60 .\) However, from your physics lectures, you remember that no gas has such a \(\gamma\) value. However, you also remember that mixing monatomic and diatomic gases can yield a gas with such a \(\gamma\) value. Determine the fraction of diatomic molecules a mixture has to have to obtain this value.