Question

The root mean square velocity of the molecules in a sample of helium is \((5 / 7)^{\text {th }}\) that of the molecules in a sample of hydrogen. If the temperature of hydrogen sample is \(0^{\circ} \mathrm{C}\), then the temperature of the helium sample is about (A) \(273^{\circ} \mathrm{C}\) (B) \(0^{\circ} \mathrm{C}\) (C) \(0^{\circ} \mathrm{K}\) (D) \(100^{\circ} \mathrm{C}\)

Short answer

The temperature of the helium sample is approximately \(273^{\circ}\mathrm{C}\).

Step-by-step solution

Write down the given information and equation

The root mean square velocity of helium is \((5/7)\) that of hydrogen. The temperature of the hydrogen sample is \(0^{\circ} \mathrm{C}\) (or \(273 \, \mathrm{K}\) in Kelvin). We need to find the temperature of the helium sample. The root mean square velocity (VRMS) formula for an ideal gas is given by: \[V_{rms} = \sqrt{\frac{3RT}{M}}\] where \(V_{rms}\) is the root mean square velocity, \(R\) is the ideal gas constant (\(8.314 \, J/mol \, K\)), \(T\) is the temperature, and \(M\) is the molar mass of the gas.

Set up the equation relating the velocities of helium and hydrogen

We can express the root mean square velocity ratio in terms of the temperatures and molar masses of helium and hydrogen: \[\frac{V_{rms}(He)}{V_{rms}(H_{2})} = \frac{5}{7}\] Now, substitute the root mean square velocity formula for the helium and hydrogen: \[\frac{\sqrt{\frac{3R T_{He}}{M_{He}}}}{\sqrt{\frac{3R T_{H_{2}}}{M_{H_{2}}}}} = \frac{5}{7}\]

Simplify and solve for the helium temperature

We want to find \(T_{He}\), so we can square both sides of the equation and then cross-multiply: \[\frac{3R T_{He}}{M_{He}} \times \frac{M_{H_{2}}}{3R T_{H_{2}}} = \frac{25}{49}\] Now, we can cancel out the \(3R\) terms and rearrange to solve for \(T_{He}\): \[T_{He} = \frac{25 M_{He} M_{H_{2}} T_{H_{2}}}{49 M_{H_{2}}}\] The molar masses of helium and hydrogen are \(4 \, g/mol\) and \(2 \, g/mol\), respectively. The temperature of hydrogen is \(273 \, K\). Substitute the values and solve for \(T_{He}\): \[T_{He} = \frac{25 \times 4 \times 2 \times 273}{49 \times 2}\]

Calculate the temperature of helium

Now, we can calculate the value of \(T_{He}\): \[T_{He} = \frac{25 \times 4 \times 2 \times 273}{49 \times 2} = 273 \times 2\] \[T_{He} = 546 \, K\] To convert the temperature back to Celsius, we subtract 273: \[T_{He} = 546 - 273 = 273^{\circ}\mathrm{C}\] Thus, the temperature of the helium sample is approximately \(273^{\circ}\mathrm{C}\). The correct answer is (A).

Key concepts

Root Mean Square Velocity

Root Mean Square Velocity, often abbreviated as RMS velocity, is a crucial concept in the study of gases. It provides a measure of the speed of particles within an ideal gas. This is especially useful in determining how gas behaves under different conditions. The formula to calculate this velocity in a sample of gas is:

• \(V_{rms} = \sqrt{\frac{3RT}{M}}\)

Here,

• \(V_{rms}\) is the root mean square velocity.
• \(R\) is the ideal gas constant (8.314 J/mol K).
• \(T\) is the temperature in Kelvin.
• \(M\) is the molar mass of the gas in kg/mol.

This equation tells us that the speed of gas particles increases with temperature and decreases with greater molar mass. Understanding RMS velocity helps predict molecular behavior and interactions in gases.

Ideal Gas Law

The Ideal Gas Law is a fundamental principle that describes the behavior of an ideal gas in terms of pressure, volume, temperature, and the number of moles. It is a powerful tool in thermodynamics and can be expressed as:

• \(PV = nRT\)

where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles of gas.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature in Kelvin.

This law helps predict how a change in one of these properties affects the others, assuming the gas behaves ideally. It is particularly useful as a foundation for understanding gas behavior and conditions in various thermodynamic processes.

Temperature Conversion

Temperature conversion is an essential aspect of thermodynamics, as calculations often require temperatures to be in a specific unit, usually Kelvin. The conversion from degrees Celsius to Kelvin is straightforward:

• \(T_{K} = T_{°C} + 273.15\)

This conversion ensures accurate calculations when working with the Ideal Gas Law or RMS velocity, as these equations require temperature in Kelvin. For example, a temperature of 0°C is equivalent to 273.15 K. Understanding how to convert temperature units quickly is crucial in solving thermodynamic problems accurately.

Molar Mass

Molar mass is a critical component in the study of gases. It is defined as the mass of one mole of a substance and is expressed in grams per mole (g/mol). For gases, it influences both the root mean square velocity and the application of the Ideal Gas Law.

• The molar mass of helium is 4 g/mol, and that of hydrogen is 2 g/mol.

Knowing the molar mass of gases helps in calculating their velocities and behavior under different thermal conditions.

• Higher molar mass means particles move more slowly compared to lighter particles at the same temperature.
• It is crucial for determining densities and for converting moles to grams in calculations.

Molar mass plays a vital role in the practical application of gas laws in chemistry and physics.