An element in its solid phase has mass density 1750 kg/m 3 and number density 4.39×10 28 atoms/m 3 . What is the element's atomic mass number?

Hey, everyone. So this problem is dealing with atomic mass numbers. Let's see what it's asking us. We have a spacecraft on a mission to explore a distant planet and that spacecraft discovers a solid mineral with a density of 2300 kg per meter cubed. The analysis reveals that the minerals number density is 6.25 times 10 to the 28 atoms per cubic meter. Find the atomic mass number of the element comprising this mineral. Our multiple choice answers here are a 28 B 26 C 24 and D 22. So the key here is to recall that the number of atoms N is equal to the equation M are mass divided by capital M or molar mass. And in turn that molar mass is equal to our atomic mass number or a multiplied by U. So we're going to use the information in the problem to first solve for our molar mass. And then we'll be able to solve for our atomic mass number. So if we rearrange this equation, this first equation to isolate molar mass, we have molar mass is equal to lowercase M mass divided by nr a number of atoms. In turn, we can recall that mass is equal to our density multiplied by our volume. So we can plug that in to our equation. So we'll have molar masses equal to row V divided by N. Now, looking at the values from the problem, we are told that this material has a mass density density of 2300 kg per cubic meters and the minerals number density. So number density is N divided by the. So that is our number over our volume and that is 6.25 times 10 to the 28 atoms per cubic meter. And so when we plug those in to our molar mass equation, sorry, let's get the right color there. So our Muller mass equation is going to be 2300 kg per meter cubed divided by 6.25 times 10 to the 28 Adams per meter cubed. And that leaves us with a molar mass of 3.68 times 10 to the negative 26 kg per atom. So now we can solve for our atomic mass number. That's a is equal to mass divided by you if we rearrange that second equation and we came up with our molar mass 3.68 times 10 to the negative 26 kg per atom. And then we can recall you is a constant 1.667 times 10 to the negative 27 kg. Which gives us an atomic mass number of 22. And so when we look at our multiple choice answers that aligns with answer choice D so D is the correct answer for this problem. That's all we have for this one, we'll see you in the next video.

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Moles and Avogadro's Number

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Moles and Avogadro's Number

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Problem 7

Textbook Question

An element in its solid phase has mass density 1750 kg/m³ and number density 4.39×10²⁸ atoms/m³. What is the element's atomic mass number?

Verified step by step guidance

Step 1: Understand the relationship between mass density, number density, and atomic mass. The mass density (ρ) of a material is related to its number density (n) and atomic mass (mₐ) by the equation: ρ = n × mₐ.

Step 2: Rearrange the formula to solve for the atomic mass (mₐ). Using the equation ρ = n × mₐ, isolate mₐ: mₐ = ρ / n.

Step 3: Substitute the given values into the formula. The mass density (ρ) is 1750 kg/m³, and the number density (n) is 4.39 × 10^28 atoms/m³. Ensure the units are consistent.

Step 4: Convert the atomic mass from kilograms to atomic mass units (u). Recall that 1 atomic mass unit (u) is equal to 1.660539 × 10^-27 kg. Use the conversion factor: mₐ (in u) = mₐ (in kg) / (1.660539 × 10^-27).

Step 5: Interpret the result. The atomic mass number is approximately the atomic mass in atomic mass units (u), rounded to the nearest whole number, as it represents the number of protons and neutrons in the nucleus of the element.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mass Density

Mass density is defined as the mass of a substance per unit volume, typically expressed in kilograms per cubic meter (kg/m³). It provides insight into how compact the matter is within a given volume. In this question, the mass density of the solid phase of the element is given as 1750 kg/m³, which is crucial for calculating the atomic mass number.

Number Density

Number density refers to the number of particles (atoms, molecules, etc.) per unit volume, usually expressed in particles per cubic meter (m⁻³). In this case, the number density of the element is provided as 4.39×10²⁸ atoms/m³. This concept is essential for relating the mass density to the atomic mass number, as it helps determine how many atoms are present in a specific volume.

Atomic Mass Number

The atomic mass number (A) is the total number of protons and neutrons in an atom's nucleus, which determines the mass of the atom. It is a key factor in calculating the mass of a mole of atoms and is related to both mass density and number density. By using the provided mass density and number density, one can derive the atomic mass number through the relationship between these quantities.

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