Problem 27 - College Physics

Question

You are designing a space station and want to get some idea about how large it should be to provide adequate air for the astronauts. Normally, the air is replenished, but for security, you decide that there should be enough to last for 2 weeks in case of a malfunction. (a) Estimate how many cubic meters of air an average person breathes in 2 weeks. A typical human breathes about \(1 / 2 \mathrm{~L}\) per breath. (b) If the space station is to be spherical, what should be its diameter to contain all the air you calculated in part (a)?

Step-by-Step Solution

Answer

The space station requires \(2352\,m^3\) of air, and a diameter of \(16.5\,m\).

Step 1: Determine the Volume of Air per Breah

A typical human breathes about \( \frac{1}{2} \) liters per breath. Since there are 1000 liters in one cubic meter, the volume of air per breath in cubic meters is \( \frac{1}{2} / 1000 = 0.0005 \) cubic meters.

Step 2: Calculate the Number of Breaths per Day

An average person takes about 12 breaths per minute. That is \( 12 \times 60 \) breaths per hour, and \( 12 \times 60 \times 24 \) breaths per day.

Step 3: Calculate Air Consumption for Two Weeks

There are 14 days in 2 weeks. Therefore, the total number of breaths in two weeks is \( 12 \times 60 \times 24 \times 14 \). Multiply this by the volume of air per breath (from Step 1) to get the total volume of air needed in cubic meters.

Step 4: Calculate the Necessary Volume of the Space Station

Using the total volume from Step 3, we set this equal to the volume of a sphere to solve for the radius using the formula \( V = \frac{4}{3} \pi r^3 \).

Step 5: Solve for the Diameter of the Sphere

Once we have \( r \), the radius of the sphere, we can find the diameter by multiplying \( r \) by 2.

Key Concepts

Air Consumption Calculation

Understanding how much air is consumed by humans over a specific period is crucial when designing a space station. For this, we first need to establish the air volume that an average person inhales with each breath. Typically, a person takes in about 0.5 liters of air per breath. To convert this to cubic meters, which is a more practical volume measurement for large spaces, we divide by 1000 (since 1000 liters equal 1 cubic meter).
The next step is to determine the total number of breaths a person takes in two weeks. Given that an average person breathes 12 times per minute, we can calculate the breaths per day by multiplying the number of breaths per minute (12) by the number of minutes in an hour (60), and then by the number of hours in a day (24). For two weeks, we multiply this daily figure by 14 days.
Finally, to find the total air consumption, we multiply the total number of breaths by the air volume per breath (0.0005 cubic meters). This provides the total air volume required per person for two weeks, an essential value for ensuring that the space station's design meets safety and health guidelines.

Spherical Geometry

Designing the space station in a spherical shape has its benefits, primarily because a sphere has the smallest surface area for a given volume, minimizing material use and maximizing strength. To calculate the required size of this sphere to contain the needed air, we start with the volume formula for a sphere: \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius.
In our exercise, the volume \( V \) is the amount of air calculated to last for two weeks. Solving for \( r \), the radius, involves rearranging the formula to \( r = \left(\frac{3V}{4\pi}\right)^{1/3} \). Once \( r \) is found, the diameter of the sphere can simply be calculated as \( 2r \).
By ensuring the sphere meets this required diameter, we guarantee that it can safely enclose all necessary air for its occupants, maintaining both structural integrity and efficiency.

Human Respiration

Human respiration is the process by which we take in air, extract the oxygen our bodies need, and expel carbon dioxide. On average, each breath has a volume of about 0.5 liters – just enough to fill our lungs without discomfort. When considering respiration for long-term living situations such as a space station, it's important to be mindful of both the volume of air per breath and the breath rate.
Our breath rate, roughly 12 breaths per minute, can vary based on activity and environmental conditions. However, for design purposes, we use the average rate to estimate air requirements. Over the course of two weeks, this breathing pattern adds up significantly, highlighting the necessity of having enough stored breathable air in environments where replenishment could be at risk.
Understanding human respiration not only ensures astronaut safety but also provides a baseline for life-support systems needed for sustaining long-term human presence in outer space.

Volume Calculation

Volume calculation is a critical part of designing a space station, particularly when determining how much space is needed for storage or habitation. In the context of air storage, it helps us ensure that there is enough room to contain the necessary amount of breathable air.
For a space with a defined geometric shape, such as a sphere, its volume is computed using mathematical formulas. In our case, the volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \). After computing the required air volume from human respiration data, this formula is utilized to derive \( r \), the sphere's radius.
This calculation not only aids in establishing the optimal size of the space station but also helps us evaluate the efficiency and feasibility of different design choices. Accurately assessing the volume ensures resources are properly allocated for life-support and other critical supplies.