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Chapter 8: Problem 29

Do all the molecules in a 1 -mole sample of \(\mathrm{CH}_{4}(g)\) have the same kinetic energy at 273 K? Do all molecules in a I-mole sample of \(\mathrm{N}_{2}(g)\) have the same velocity at \(546 \mathrm{K} ?\) Explain.

Short Answer

Expert verified
In conclusion, all molecules in a 1-mole sample of CH4(g) do not have the same kinetic energy at 273 K, and all molecules in a 1-mole sample of N2(g) do not have the same velocity at 546 K. This is because the distribution of velocities in both cases follows the Maxwell-Boltzmann distribution, which demonstrates that gas particles have a range of velocities rather than a single, uniform velocity.

Step by step solution

01

The kinetic energy of a single gas molecule is given by the equation: \[KE = \frac{1}{2}mv^2\] where \(KE\) is the kinetic energy, \(m\) is the mass of the molecule, and \(v\) is the velocity of the molecule. The root-mean-square velocity (v_rms) is a measure used to define the average velocity of gas molecules and is defined as: \[v_\text{rms} = \sqrt{ \frac{3kT}{m} }\] where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of the molecule. #Step 2: Explain the Maxwell-Boltzmann distribution#

The Maxwell-Boltzmann distribution is a probability distribution that describes the distribution of molecular speeds in an ideal gas. It shows that particles in a gas do not all have the same velocity; rather, they have a range of velocities that follow a bell-shaped curve. This means that even though there is an average velocity (v_rms), individual molecules in the gas may have velocities higher or lower than the v_rms. #Step 3: Determine the kinetic energy of CH4 molecules at 273 K#
02

Since the kinetic energies of the individual molecules depend on their velocities, and we know that the molecules in the CH4 gas have a range of velocities due to the Maxwell-Boltzmann distribution, we can conclude that all CH4 molecules do not have the same kinetic energy at 273 K. #Step 4: Determine and compare velocities of N2 molecules at 546 K#

Similarly, the velocities of the N2 molecules at 546 K will be distributed according to the Maxwell-Boltzmann distribution. As a result, we can conclude that not all molecules in a 1-mole sample of N2(g) have the same velocity at 546 K. In conclusion, all molecules in a 1-mole sample of CH4(g) do not have the same kinetic energy at 273 K, and all molecules in a 1-mole sample of N2(g) do not have the same velocity at 546 K. The distribution of velocities in both cases follows the Maxwell-Boltzmann distribution, which demonstrates that gas particles have a range of velocities rather than a single, uniform velocity.

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Most popular questions from this chapter

A sealed balloon is filled with \(1.00 \mathrm{L}\) helium at \(23^{\circ} \mathrm{C}\) and 1.00 atm. The balloon rises to a point in the atmosphere where the pressure is \(220 .\) torr and the temperature is \(-31^{\circ} \mathrm{C}\). What is the change in volume of the balloon as it ascends from \(1.00\) atm to a pressure of \(220 .\) torr?

Without looking at a table of values, which of the following gases would you expect to have the largest value of the van der Waals constant \(b: \mathrm{H}_{2}, \mathrm{N}_{2}, \mathrm{CH}_{4}, \mathrm{C}_{2} \mathrm{H}_{6},\) or \(\mathrm{C}_{3} \mathrm{H}_{8} ?\)

Consider a \(1.0\) -\(\mathrm{L}\) container of neon gas at STP. Will the average kinetic energy, average velocity, and frequency of collisions of gas molecules with the walls of the container increase, decrease, or remain the same under each of the following conditions? a. The temperature is increased to \(100^{\circ} \mathrm{C}\) b. The temperature is decreased to \(-50^{\circ} \mathrm{C}\) c. The volume is decreased to \(0.5 \mathrm{L}\) d. The number of moles of neon is doubled.

Calculate \(w\) and \(\Delta E\) when \(1\) mole of a liquid is vaporized at its boiling point \(\left(80 .^{\circ} \mathrm{C}\right)\) and \(1.00\) atm pressure. \(\Delta H\) for the vaporization of the liquid is \(30.7 \mathrm{kJ} / \mathrm{mol}\) at \(80 .^{\circ} \mathrm{C}\). Assume the volume of \(1\) mole of liquid is negligible as compared to the volume of \(1\) mole of gas at \(80 .^{\circ} \mathrm{C}\) and \(1.00\) atm.

Which of the following statements is(are) true? a. If the number of moles of a gas is doubled, the volume will double, assuming the pressure and temperature of the gas remain constant. b. If the temperature of a gas increases from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\) the volume of the gas would double, assuming that the pressure and the number of moles of gas remain constant. c. The device that measures atmospheric pressure is called a barometer. d. If the volume of a gas decreases by one half, then the pressure would double, assuming that the number of moles and the temperature of the gas remain constant.
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