Calculating the Number of Particles in Your Room

[EV33]

The EPA has proposed a new standard for microparticulates in air: for particles up to 2.5 µm in diameter, the maximum allowable amountin 50 µg/m^3. If your 9.3 ft x 9.25 ft x 11.3 ft dorm room just meets the new EPA standard, how many of these particles are in your room?
How many of these particles are in each 0.500 L breath you take?
(Assume the particles are spheres of 2.5 µm diameter and made primarily of soot, a form of carbon with a density of 2.5 g/cm^3.)



One of my friends got this problem for homework and I was trying to help her out. If someone could tell me if these are the right steps that would be awesome. Just for the first problem.



1. Find the volume of one particle.
2. Multiply the Volume by the density
3. The answer of that would give you the mass of one particle. Convert from grams to micrograms.
4. 50 micrograms divided by the mass of one particle.
6. That answer tells you the amount of particles per cubic cm
7. Multiply the amount of particles per cubic cm by the volume of the room in meters cubed.


does that sound right? Or am I way off?

 

[Borek]

In general you are OK, just watch your units, you mixed ccm and m³.

 

[Blueshift5]

Your approach is mostly correct, but there are a few things to consider. Here is a step-by-step guide to calculating the number of particles in your room:

1. Find the volume of your room in cubic meters. To do this, convert the dimensions from feet to meters (1 ft = 0.3048 m) and then multiply them together. In this case, the volume of your room is approximately 2.74 m^3.

2. Calculate the maximum allowable amount of particles in your room based on the EPA standard. This is given in units of micrograms per cubic meter, so we need to convert the room volume to cubic meters (1 m^3 = 1000000 cm^3) and then multiply by the maximum allowable amount (50 µg/m^3). This gives us a maximum of 137,000 particles in your room (2.74 m^3 * 50 µg/m^3 * 1000000 cm^3/m^3).

3. Calculate the volume of one particle. The volume of a sphere is given by V = (4/3)πr^3, where r is the radius of the sphere. In this case, the diameter of the particles is given as 2.5 µm, so the radius is 1.25 µm or 1.25 x 10^-6 m. Plugging this into the formula, we get a volume of approximately 6.54 x 10^-20 m^3.

4. Calculate the mass of one particle. We can use the density of soot (2.5 g/cm^3) to convert the volume of one particle to its mass. This gives us a mass of approximately 1.64 x 10^-19 g or 1.64 x 10^-16 µg.

5. Calculate the number of particles in each breath. The average breath volume is approximately 0.5 L or 0.0005 m^3. To calculate the number of particles in this volume, we divide the breath volume by the volume of one particle. This gives us approximately 7.64 x 10^14 particles per breath (0.0005 m^3 / 6.54 x 10^-20 m^3).

6. Calculate the number of particles in your room. To find the total number of particles in your room, we can divide the maximum allowable amount of particles (