Problem 51 - College Physics

Question

The critical density of the universe is \(5.8 \times 10^{-27} \mathrm{kg} / \mathrm{m}^{3}\) . (a) Assuming that the universe is all hydrogen, express the critical density in the number of \(\mathrm{H}\) atoms per cubic meter. (b) If the density of the universe is equal to the critical density, how many atoms, on the average, would you expect to find in a room of dimensions 4 \(\mathrm{m} \times 7 \mathrm{m} \times 3 \mathrm{m} ?\) (c) Compare your answer in part (b) with the number of atoms you would find in this room under normal conditions on the earth.

Step-by-Step Solution

Answer

(a) Approximately 3.47 hydrogen atoms/m³. (b) About 291 atoms in the room. (c) Contrast this with the approx. 2.5 x 10²⁵ atoms under normal Earth conditions.

Step 1: Determine Mass of a Hydrogen Atom

The mass of a single hydrogen atom can be approximated by the mass of a proton. This is approximately \(1.67 \times 10^{-27} \mathrm{kg}\).

Step 2: Calculate the Number of H Atoms per Cubic Meter

To find the number of hydrogen atoms per cubic meter, divide the critical density by the mass of a single hydrogen atom:\[\frac{5.8 \times 10^{-27} \mathrm{kg/m}^3}{1.67 \times 10^{-27} \mathrm{kg}} \approx 3.47 \text{ atoms/m}^3.\]

Step 3: Calculate the Volume of the Room

The volume of a room with dimensions \(4 \mathrm{m} \times 7 \mathrm{m} \times 3 \mathrm{m}\) is found by multiplying these dimensions:\[ 4 \mathrm{m} \times 7 \mathrm{m} \times 3 \mathrm{m} = 84 \mathrm{m}^3.\]

Step 4: Calculate Number of Atoms in the Room

Multiply the number of hydrogen atoms per cubic meter by the volume of the room to find the total number of hydrogen atoms in the room:\[3.47 \text{ atoms/m}^3 \times 84 \mathrm{m}^3 \approx 291 \text{ hydrogen atoms}.\]

Step 5: Compare with Earth's Conditions

Under normal Earth conditions, a room this size can contain approximately \(2.5 \times 10^{25}\) atoms, based on air density and composition. Compare this number to the number calculated under the assumption of critical density.

Key Concepts

Hydrogen Atom Mass

To understand the critical density of the universe in terms of hydrogen atoms, we first need to grasp the mass of a single hydrogen atom. A hydrogen atom consists mainly of a single proton. The mass of a proton is approximately \(1.67 \times 10^{-27} \text{ kg}\). This tiny mass might seem insignificant, but it plays a crucial role in universe density calculations. The mass of a hydrogen atom is pivotal because it provides a baseline to calculate how many such atoms fit in a given mass density, such as the critical density of the universe. By using this approximate mass, we can convert mass density into the number of atoms, which is a more intuitive measurement for many calculations.

Universe Density Calculations

Calculating universe density, particularly the critical density, involves understanding how many hydrogen atoms fit in a cubic meter. The critical density of the universe is the theoretical density needed for the universe to remain in a delicate balance between expansion and gravitational collapse. Given as \(5.8 \times 10^{-27} \text{ kg/m}^3\), this density is quite low. By dividing this critical density by the mass of a hydrogen atom, \(1.67 \times 10^{-27} \text{ kg}\), we approximate that there are about \(3.47\) hydrogen atoms per cubic meter. This conversion helps conceptualize the sparseness of matter spread throughout the universe, making it easier to understand vast and complex cosmological concepts in understandable physical terms.

Comparison with Earth's Conditions

To comprehend how vastly different the universe's density is compared to Earth, let's consider a room with dimensions \(4 \text{ m} \times 7 \text{ m} \times 3 \text{ m}\). If the universe's density matched the critical density, you'd find approximately \(291\) hydrogen atoms in such a room. In stark comparison, under normal Earth conditions, this same room would contain about \(2.5 \times 10^{25}\) atoms. Earth, with its dense atmosphere and material composition, shows how dense a place it is compared to the expansive and less dense universe. This comparison highlights the immense difference in particle distribution, driving home how unique Earth's conditions are compared to the universe's baseline state. Understanding these numbers underscores the scale and nature of the matter we encounter daily versus the vastness of the cosmos.