Question 906: Thermodynamics & KTG

Question

Equal volumes of two gases, having their densíties in the ratio of $$1: 16$$ exert equal pressures on the walls of two containers. The ratio of their rms speads ($$\mathrm{C}_1: \mathrm{C}_2)$$ is

Accepted Answer

To solve this problem, we need to understand that the root-mean-square (rms) speed of gas molecules is related to the temperature and the mass of the gas molecules. The formula for the rms speed of gas is given by:
$$ \mathrm{C} = \sqrt{\frac{3kT}{m}} $$
where:
$$\mathrm{C}$$ is the rms speed,
$$k$$ is the Boltzmann constant,
$$T$$ is the temperature, and
$$m$$ is the mass of the gas molecule.
We are given that the gases have equal volumes and exert equal pressures. From the ideal gas law:
$$ PV = nRT $$
where $$P$$ is the pressure, $$V$$ is the volume, $$n$$ is the number of moles, $$R$$ is the ideal gas constant, and $$T$$ is the temperature. If the pressures are equal and the volumes are equal, then it implies that the product $$nT$$ must be the same for both gases.
In this problem, the temperatures are the same because the pressures and volumes are the same for both gases. Given that the densities are in the ratio $$1:16$$, we can write:
$$ \frac{\rho_1}{\rho_2} = \frac{1}{16} $$
where $$\rho = \frac{m}{V}$$, and since the volumes are the same, the ratio of the masses of the individual molecules will be:
$$ \frac{m_1}{m_2} = \frac{1}{16} $$
Now, substituting this ratio back into the rms speed formula, we have:
$$ \mathrm{C}_1 = \sqrt{\frac{3kT}{m_1}} $$
$$ \mathrm{C}_2 = \sqrt{\frac{3kT}{m_2}} $$
Thus, the ratio of their rms speeds is:
$$ \frac{\mathrm{C}_1}{\mathrm{C}_2} = \sqrt{\frac{m_2}{m_1}} = \sqrt{\frac{16}{1}} = 4 $$
Therefore, the ratio of the rms speeds is $$\mathrm{4:1}$$.
The correct answer is
Option B: 4:1
.

Answer

$$1: 4$$

Answer

$$4: 1$$

Answer

$$8: 1$$

Answer

$$1: 8$$

Thermodynamics & KTG

Equal volumes of two gases, having their densíties in the ratio of $1: 16 exert equal pressures on the walls of two containers. The ratio of their rms speads (\mathrm{C}_1: \mathrm{C}_2)$ is

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