Question

The mass composition of universe may be given as \(90 \% \mathrm{H}_{2}\) and \(10 \%\) He. The average molecular mass of universe should be (a) \(2.20\) (b) \(2.10\) (c) \(3.80\) (d) \(3.64\)

Short answer

The average molecular mass of the universe is 2.20.

Step-by-step solution

Identify the molecular masses of hydrogen and helium

The molecular mass of hydrogen (H2) is 2 amu, since it consists of two hydrogen atoms each with a mass of approximately 1 amu. The atomic mass of helium (He) is approximately 4 amu.

Calculate the weighted average molecular mass

To find the average molecular mass of the universe, use the given mass composition as weights. Multiply the molecular mass of hydrogen by its percentage and the atomic mass of helium by its percentage, and then sum the results. Formula: (Molecular mass of H2 * Percentage of H2) + (Atomic mass of He * Percentage of He) Calculation: (2 amu * 90/100) + (4 amu * 10/100)

Simplify and find the average molecular mass

Now, simplify the calculation and obtain the result. (2 * 0.9) + (4 * 0.1) = 1.8 + 0.4 = 2.2

Key concepts

Mass Composition of the Universe

Understanding the mass composition of the universe is akin to creating a recipe for a cosmic mix. This involves determining the percentage of each element or molecule that collectively makes up the vast expanse of space. Generally, the universe is primarily composed of hydrogen (H₂) and helium (He), with hydrogen being the most abundant element.

The mass composition provided in the exercise explains that the universe consists of 90% hydrogen and 10% helium. But why are these percentages important? They play a crucial role when we calculate the average molecular mass of the universe. This is because the physical properties of the cosmos, such as its density and gravitational behavior, depend significantly on the mass and distribution of these elements.

By understanding the mass composition, astronomers and scientists can infer a great deal about the formation, evolution, and current state of the universe. It also aids in determining how galaxies come together, how stars form and evolve, and the way in which various celestial phenomena occur.

Molecular Mass Calculation

Molecular mass calculation is fundamental in both chemistry and astrophysics, serving as a lens through which we can evaluate the nature of substances. In straightforward terms, the molecular mass of a substance indicates the mass of a single molecule of that substance, usually expressed in atomic mass units (amu).

In our case, the molecular mass of hydrogen (H₂), which is a molecule made up of two hydrogen atoms, amounts to 2 amu, since each hydrogen atom approximately equals 1 amu. Helium (He), on the other hand, is an element, and its atomic mass is about 4 amu.

When we want to calculate the average molecular mass of a mixture, such as the universe's primarily hydrogen and helium composition, we can't simply take an average. Instead, we must consider each component's mass and its relative abundance, akin to calculating the mean weight of a group where members have significantly different weights and numbers.

Weighted Average

The concept of a weighted average is essential when different values in a data set contribute more or less significantly to the final outcome. Unlike a simple average, which treats all values equally, a weighted average multiplies each number by a designated weight before summing them up and dividing by the sum of the weights.

In the context of computing the average molecular mass of the universe, we use the mass percentages of hydrogen and helium as the weights. This gives us a more accurate representation of the universe's makeup because we're acknowledging that hydrogen, being 90% of the universe, should contribute more to the average than helium.

The calculation becomes a two-step process where we first multiply the molecular mass of each element by its respective weight (percentage), then sum these products to find the weighted average. This value reflects the overall heft of one molecule on average, given the proportions of hydrogen and helium that make up our cosmos.