Question

Ordinary hydrogen is a mixture of the isotopes II and D. In which of the following propcrtics would the two gases \(\mathrm{II}_{2}\) and \(\mathrm{D}_{2}\) differ? (1) Number of molecules present in a given volume at STP (2) Rate of diffusion under the same conditions of temperature and pressure (3) Colour (4) Number of orbital clectrons

Short answer

\(\text{H}_2\) and \(\text{D}_2\) differ in their rate of diffusion under the same conditions of temperature and pressure.

Step-by-step solution

Understanding Isotopes

Isotopes are atoms of the same element that have different numbers of neutrons. In this exercise, the isotopes II (Hydrogen-1) and D (Deuterium, Hydrogen-2) differ by one neutron.

Evaluating Property 1: Number of Molecules Present in a Given Volume at STP

Under standard temperature and pressure (STP), 1 mole of any gas occupies 22.4 liters. This means that both \(\text{H}_2\) and \(\text{D}_2\) will have the same number of molecules in a given volume at STP.

Evaluating Property 2: Rate of Diffusion

Graham's law of diffusion states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Since Deuterium (\text{D}_2) has a higher molar mass than Hydrogen (\text{H}_2), \(\text{H}_2\) will diffuse faster than \(\text{D}_2\).

Evaluating Property 3: Colour

Both \(\text{H}_2\) and \(\text{D}_2\) are colourless gases. Hence, they do not differ in colour.

Evaluating Property 4: Number of Orbital Electrons

Both isotopes, Hydrogen-1 and Deuterium, have the same number of protons and electrons. Therefore, they do not differ in the number of orbital electrons.

Conclusion

Based on the analysis of the properties, \(\text{H}_2\) and \(\text{D}_2\) differ only in their rate of diffusion under the same conditions of temperature and pressure.

Key concepts

Graham's Law of Diffusion

Graham's law of diffusion helps us understand how gases spread or mix with each other. It tells us that lighter gases diffuse faster than heavier ones. This is because the rate at which a gas diffuses is inversely proportional to the square root of its molar mass. So, if you have two gases, the one with the lower molar mass will move quicker.
For example, hydrogen gas \(\text{H}_2\) and deuterium gas \(\text{D}_2\) differ in mass. \(\text{D}_2\) is heavier due to the presence of an extra neutron in each molecule. Thus, \(\text{H}_2\) will diffuse faster than \(\text{D}_2\). This idea can be represented using the equation for Graham's law: \[ \frac{\text{Rate of diffusion of gas 1}}{\text{Rate of diffusion of gas 2}} = \sqrt{\frac{\text{Molar mass of gas 2}}{\text{Molar mass of gas 1}}} \]
In this way, we can predict how quickly a gas will diffuse compared to another.

Standard Temperature and Pressure (STP)

Understanding STP is essential when studying the properties of gases. STP stands for Standard Temperature and Pressure, and it is used as a reference point in gas calculations.
Standard temperature is 0 degrees Celsius (273.15 Kelvin), and standard pressure is 1 atmosphere (atm). At these conditions, one mole of any gas occupies a volume of 22.4 liters.
This means that hydrogen \(\text{H}_2\) and deuterium \(\text{D}_2\) at STP will have the same number of molecules if they have the same volume. The formula \[ \text{PV} = \text{nRT} \] denotes the ideal gas law, where:

• \(\text{P}\) = pressure
• \(\text{V}\) = volume
• \(\text{n}\) = number of moles
• \(\text{R}\) = universal gas constant
• \(\text{T}\) = temperature

This relationship helps us see that under the same conditions of STP, the volume occupied by a gas is directly proportional to the number of moles of gas, regardless of its type.

Molecular Properties of Gases

Gases have unique molecular properties that define their behavior. Some of these properties include:

• Diffusion: The process in which gas molecules spread out to fill the available space. Graham's law, as discussed earlier, is a fundamental principle of diffusion.
• Compressibility: Gases can be compressed into smaller volumes when pressure is applied due to the significant amounts of empty space between molecules.
• Expansion: Gases expand to fill their containers, so if you increase the volume of a container, the gas will spread to fill the entire space.
• Low Density: Gases generally have much lower densities compared to liquids and solids because their molecules are far apart.
• Elastic Collisions: Gas molecules collide with each other and the walls of their container without losing energy. This is why pressure is exerted by gases.
• Temperature and Pressure Relationship: According to the ideal gas law, at a constant volume, the pressure of a gas is directly proportional to its temperature, and at a constant pressure, the volume is directly proportional to its temperature.

Each of these properties is significant in understanding how gases behave under various conditions, and they help explain why gases like \(\text{H}_2\) and \(\text{D}_2\) act differently under the same conditions.