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Chapter 5: Problem 33

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
No, all the molecules in a 1-mole sample of CH4(g) at 273 K do not have the same kinetic energy, and all molecules in a 1-mole sample of N2(g) at 546 K do not have the same velocity. This is because of the Maxwell-Boltzmann distribution, which describes the probability distribution of molecular speeds in a gas. Although the molecules have the same average kinetic energy at a constant temperature, they have a range of individual velocities and kinetic energies due to this distribution.

Step by step solution

01

Recall the Maxwell-Boltzmann Distribution

Maxwell-Boltzmann distribution is a statistical distribution that describes the probability of a molecule in a gas having a particular speed. According to this distribution, gas molecules have a range of velocities, with the average molecular velocity determined by the temperature of the system.
02

Determine the kinetic energy of molecules

The kinetic energy (KE) of a molecule in a gas is given by the equation: KE = \(\frac{1}{2}mv^{2}\) where m is the mass of the molecule, and v is its velocity. At a constant temperature, the average kinetic energy of the molecules in a gas will be the same regardless of the type of gas. Therefore, if two gases are both at 273 K, their molecules will have the same average kinetic energy.
03

Answer if all CH4 molecules have the same kinetic energy at 273 K

Although all the CH4 molecules in the sample will have the same average kinetic energy, they will not all have the same individual kinetic energies. This is because, as mentioned above, gas molecules have a distribution of velocities, and their kinetic energy is directly related to their velocity according to the equation in Step 2.
04

Determine the relationship between molecular mass and velocity

As mentioned earlier, the kinetic energy of a molecule is given by KE = \(\frac{1}{2}mv^{2}\). Since the average kinetic energy is the same for gases at the same temperature, we can set up the following equation for CH4 and N2: \(\frac{1}{2}mv_{\text{CH4}}^2\) = \(\frac{1}{2}mv_{\text{N2}}^2\) Where m is the mass and v is the velocity of the respective molecules. We see that the mass of the molecule directly affects its velocity. Specifically, for the same kinetic energy, a lighter molecule (CH4) will have a higher velocity compared to a heavier molecule (N2).
05

Answer if all N2 molecules have the same velocity at 546 K

Like CH4 molecules, all N2 molecules will have the same average kinetic energy but will not have the same individual velocities because of the Maxwell-Boltzmann distribution. The molecules will have a variety of velocities owing to their distribution, and the effect of temperature on their motion. In conclusion, all CH4 molecules in a 1-mole sample at 273 K temperature will have the same average kinetic energy; however, they will not have the same individual kinetic energies. Similarly, all N2 molecules in a 1-mole sample at 546 K will not have the same individual velocities due to the Maxwell-Boltzmann distribution.

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