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Chapter 5: Problem 29

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

Short Answer

Expert verified
In conclusion, not all molecules in a 1-mole sample of CH₄(g) will have the same kinetic energy at 273 K, and not all molecules in a 1-mole sample of N₂(g) will have the same velocity at 546 K. This is due to the distribution of kinetic energy and velocity of individual molecules in a gas, according to the Maxwell-Boltzmann distribution.

Step by step solution

01

Exploring kinetic energy formula for gases

For a sample of gas, the average kinetic energy of gas particles can be represented using the equation \( KE_{avg} = \frac{3}{2} kT \), where \(KE_{avg}\) is the average kinetic energy, \(k\) is the Boltzmann constant, and \(T\) is the temperature in Kelvin. But, gas particles have a distribution of kinetic energy, and not all particles possess the same kinetic energy. However, the average kinetic energy is directly proportional to the temperature of the gas in Kelvin.
02

Discussing CH₄(g) kinetic energy at 273 K

Since the average kinetic energy of molecules in a gas is directly proportional to the temperature, every molecule of the CH₄(g) will have the same average kinetic energy at 273 K. However, individual molecules will still have a distribution of kinetic energy due to their molecular motion and interactions in the gas. In summary, not all molecules in the 1-mole sample of CH₄(g) will have the same kinetic energy at 273 K.
03

Exploring the relationship between the velocity of gas molecules

To analyze the velocity of molecules in N₂(g) at 546 K, we can use the Maxwell-Boltzmann distribution, which states that the probability of a gas molecule having a certain speed is given by its mass, temperature, and the Boltzmann constant. This distribution shows that individual molecules of a gas have a range of velocities, and not all of them will possess the same velocity at a given temperature.
04

Discussing N₂(g) velocity at 546 K

In the case of the 1-mole sample of N₂(g) at 546 K, the Maxwell-Boltzmann distribution will give us a range of velocities for individual molecules. Therefore, not all molecules in the 1-mole sample of N₂(g) will have the same velocity at 546 K. In conclusion, both cases present a distribution of kinetic energy and velocity for individual molecules, indicating that not all molecules in a 1-mole sample of CH₄(g) will have the same kinetic energy at 273 K, and not all molecules in a 1-mole sample of N₂(g) will have the same velocity at 546 K.

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Most popular questions from this chapter

A 15.0-L tank is filled with \(\mathrm{H}_{2}\) to a pressure of \(2.00 \times 10^{2}\) atm. How many balloons (each \(2.00 \mathrm{~L}\) ) can be inflated to a pressure of \(1.00\) atm from the tank? Assume that there is no temperature change and that the tank cannot be emptied below \(1.00\) atm pressure.

A 20.0-L stainless steel container at \(25^{\circ} \mathrm{C}\) was charged with \(2.00\) atm of hydrogen gas and \(3.00\) atm of oxygen gas. A spark ignited the mixture, producing water. What is the pressure in the tank at \(25^{\circ} \mathrm{C} ?\) If the exact same experiment were performed, but the temperature was \(125^{\circ} \mathrm{C}\) instead of \(25^{\circ} \mathrm{C}\), what would be the pressure in the tank?

An important process for the production of acrylonitrile \(\left(\mathrm{C}_{3} \mathrm{H}_{3} \mathrm{~N}\right)\) is given by the following equation: \(2 \mathrm{C}_{3} \mathrm{H}_{6}(g)+2 \mathrm{NH}_{3}(g)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{C}_{3} \mathrm{H}_{3} \mathrm{~N}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)\) A \(150 .-\mathrm{L}\) reactor is charged to the following partial pressures at \(25^{\circ} \mathrm{C}\) : $$ \begin{aligned} P_{\mathrm{C}_{3} \mathrm{H}_{6}} &=0.500 \mathrm{MPa} \\ P_{\mathrm{NH}_{3}} &=0.800 \mathrm{MPa} \\ P_{\mathrm{O}_{2}} &=1.500 \mathrm{MPa} \end{aligned} $$ What mass of acrylonitrile can be produced from this mixture \(\left(\mathrm{MPa}=10^{6} \mathrm{~Pa}\right) ?\)

A steel cylinder contains \(150.0\) moles of argon gas at a temperature of \(25^{\circ} \mathrm{C}\) and a pressure of \(8.93 \mathrm{MPa}\). After some argon has been used, the pressure is \(2.00 \mathrm{MPa}\) at a temperature of \(19^{\circ} \mathrm{C}\). What mass of argon remains in the cylinder?

Given that a sample of air is made up of nitrogen, oxygen, and argon in the mole fractions \(0.78 \mathrm{~N}_{2}, 0.21 \mathrm{O}_{2}\), and \(0.010 \mathrm{Ar}\), what is the density of air at standard temperature and pressure?
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