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Chapter 17: Problem 6

Two different gases are at the same temperature, and both have low enough densities that they behave like ideal gases. Do their molecules have the same thermal speeds? Explain.

Short Answer

Expert verified
No, different gases at the same temperature do not have the same thermal speeds. Even if the temperature is same, the thermal speed varies due to difference in molar masses.

Step by step solution

01

Understand Basic Principles

According to kinetic theory, the kinetic energy of gas molecules is directly proportional to temperature. This is expressed as: \( K.E = \frac{3}{2} kT \), where \(K.E\) is the kinetic energy, \(k\) is Boltzmann's constant and \(T\) is the temperature in Kelvin. The root mean square speed (\(v_{rms}\)) is then given by \(v_{rms} = \sqrt{\frac{3kT}{m}}\) where \(m\) is the mass of a gas molecule. It is clear from this equation that the root mean square speed depends on the temperature and the mass of the gas molecule. Two gases of the same temperature but different molar masses will not have the same \(v_{rms}\).
02

Compare Thermal Speeds of Two Gases

Going by the formula above, even if two gases are at the same temperature, if their molar masses are different, their thermal speeds will differ. Lighter molecules will move faster, while heavier molecules will have slower thermal speeds.
03

Conclusion

Therefore, in conclusion, the molecules of the two gases do not have the same thermal speeds; since their molar mass is likely to be different. Even though their average kinetic energies might be the same due to having the same temperature, their thermal speeds vary because lighter molecules will move faster while heavier ones will move slower at the same temperature.

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Most popular questions from this chapter

The atmospheres of relatively low-mass planets like Earth don't contain much hydrogen (H \(_{2}\) ), while more massive planets like Jupiter have considerable atmospheric hydrogen. What factors might account for the difference?

According to the ideal-gas law, what should be the volume of a gas at absolute zero? Why is this result absurd?

Water's coefficient of volume expansion in the temperature range from \(0^{\circ} \mathrm{C}\) to about \(20^{\circ} \mathrm{C}\) is given approximately by \(\beta=a+b T+c T^{2},\) where \(T\) is in Celsius and \(a=-6.43 \times 10^{-50} \mathrm{C}^{-1}, b=1.70 \times 10^{-50} \mathrm{C}^{-2},\) and \(c=\) \(-2.02 \times 10^{-7 \circ} \mathrm{C}^{-3} .\) Show that water has its greatest density at approximately \(4.0^{\circ} \mathrm{C}\).

If the volume of an ideal gas is increased, must the pressure drop proportionately? Explain.

The average speed of the molecules in a gas increases with increasing temperature. What about the average velocity?
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